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Course Title: Grade: Honors Calculus, Grade 12, Level 4 12 Length of Course: One Year (5 credits) Prerequisites: Algebra, Geometry, Algebra 2, Precalculus Description: This Calculus course aims at preparing students for success in college-level Math. At the foundation of mathematics is the idea of a function. Functions express the way one variable quantity is related to another quantity. Calculus was invented to deal with the rate at which a quantity varies, particularly if that rate does not stay constant. Clearly, this course needs to begin with a thorough review of functions– particularly the properties, behavior and manipulation of important functions such as polynomial, exponential, logarithmic and trigonometric functions. Beyond a review of functions, students will have a firm understanding of the derivative and its applications as well as the integral (both definite and indefinite) and its applications. In this class we will be guided by two principles. The first is The Rule of Three which requires that every topic be presented geometrically, numerically and algebraically. The second guiding principle is The Way of Archimedes which states that formal definitions and procedures evolve from the investigation of practical problems. This course is VERY user friendly. The problems we consider come from the Social Sciences, Life Sciences and Business arenas and are generally understood to be important. This course strives to give students a proper balance between the mastery of skills and the comprehension of key concepts. With that in mind, this curriculum guide clearly defines the learning objectives for each unit in terms of the key skills and key concepts that must be mastered within each unit. Evaluation: Student performance will be measured using a variety of instructor-specific quizzes and chapter tests as well as a common departmental Midterm and Final Exam. Assessments will equally emphasize measurement of the degree to which required skills have been mastered as well as how well key concepts have been understood. Text: Calculus: Concepts & Contexts 3rd Edition, James Stewart, Thomson Brooks/Cole 2005 Reference Texts: Calculus, Deborah Hughes-Hallett, Andrew M. Gleason, et al, Wiley & Sons, 1994 Calculus: Graphical, Numerical, Algebraic, Ross Finney, Franklin Demana, et al, Pearson/Prentice Hall, 2007 Topic 1.1 Four Ways To Represent A Function Honors Calculus Grade 12 Unit 1: Functions & Models Learning Objectives: Key Definitions, Skills and Concepts What is a function? Why do we care about whether a pairing of data is a function? What are the four ways a function can be represented? What is the Vertical Line Test? What is a piecewise function? How do we recognize and note when a function has symmetry? When is a function increasing, decreasing? Skills check, ability to: Recognize a function from its graph, algebraic representation, and table Given either the table or graph of a function, be able to evaluate f(c), c a constant Given the algebraic representation of a function be able to evaluate f( x + h) State the domain and range of a function State the intervals over which a function is increasing/decreasing Determine if the graph is that of a function using the vertical line test Create a graph of a function given anecdotal data, a table or an algebraic equation Given either the graph or algebraic representation of a function, determine if its even or odd Concept check: What does it mean to say that f is a function of x? True or false: f (x) the same as f x . Explain why. Explain how a function is similar to a machine. Be able to pair math words with Input, output and rule. Be able to state the ‘golden’ rule of a function. If f is a function and f(a) = f(b), must a=b? Why or why not? Grade 12 Honors Calculus, page 2 Topic 1.2 Essential Functions Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What is a mathematical model? What is the use of a mathematical model? What are the essential functions from which we will create models? Skills check, ability to: Create the following kinds of models from data: linear, polynomial, power, rational, trigonometric, exponential, logarithmic Choose the most appropriate model based upon the data Use technology to create a model Concept check: When can we be confident in the predictions based upon modeling? What is interpolation v. extrapolation? What is the real world significance of constants we determine in modeling? (i.e. the slope and y-intercept in a linear model, the base in an exponential model) 1.3 Transformations of Functions What are dilations and translations? How are these accomplished graphically and algebraically? What is composition of functions? What is the domain of a composed function? How are reflections over the coordinate axes and y=x accomplished graphically and algebraically? Skills check, ability to: Draw the graph of f(x c), f(cx), cf(x), and f(x) c, given the graph of f Identify parent curves Evaluate f(g(c)) given rules, graphs or tables for f and g Find f(g(x)) given rules for f and g Find the domain of f(g(x)) given domains for f and g Express a complicated function as the composition of easier functions Concept check: What is the benefit of understanding about transformations of functions? What are the real world applications of composition of functions? If c > 0, why does f ( x c) shift the graph of f (x) to the right and not to the left as one might expect? Is the function f(g(x)) the same as g(f(x))? Why or why not? If f(x) is linear and g(x) is linear, must f(g(x)) also be linear? Why or why not? Grade 12 Honors Calculus, page 3 Honors Calculus Grade 12 1.4 Graphing Calculators How can the graphing calculator be used to efficiently aid in our study of functions? Skills check, ability to: Correctly graph a function Identify a correct window in order to show the portion of the function in which we are interested Correctly determine the intersection of two functions Create a customized table of values for a function Solve an equation using the graphing calculator Use the regression feature of a graphing calculator Use the features of a graphing calculator to identify domain and range of a function Concept check: 1.5 Exponential Functions What does the graph of an exponential function look like? What is its domain and range? What is e and when is it used? What distinguishes the graph of decay v growth? What is half life? Skills check, ability to: Graph exponential functions, and identify domain and range Apply transformations on exponential functions Use an exponential function as one function when composing functions Express an exponential function in y = c*a x Correctly calculate with half life problems Concept check: What distinguishes an exponential function from a linear function? Why is e called the natural base? Identify real world uses of exponential functions. State a real world question that has Total = 90 * 3 x as its answer. Grade 12 Honors Calculus, page 4 Topic 1.6 Inverse Functions & Logarithmic Functions Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What is the definition of an inverse function? Do all functions have inverse functions? How can you tell if a function will have an inverse that is also a function? What is the horizontal like test? How can you graphically construct inverses? What is a one-to-one function? How can you algebraically construct inverses? What is a logarithmic function? What is the relationship between an exponential and logarithmic function? What is a natural log? How can you use a calculator to compute logs to bases other than 10 and e? What is the change of base formula? What kinds of equations do logs allow you to solve? Skills check, ability to: Test for whether a given function is one-to-one and the existence of an inverse function (horizontal line test) Verify whether two functions are inverses Find the inverse of a one-to-one function (algebraic, graphical, and numerical method) and its corresponding domain and range Switch back and forth from logarithmic to exponential expressions. Evaluate logarithms using basic properties of logarithms Graph logarithmic functions Apply transformations of logarithmic functions (reflections, vertical translation and horizontal translation) Evaluate common logarithms Evaluate natural logarithms Find the domain of a logarithmic function Concept check: How can a function that is not one-to-one, have an inverse function? In the expression f 1 (x), is -1 an exponent? Grade 12 Honors Calculus, page 5 Topic 1.7 Parametric Curves Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What is a parametric curve? What can parametric curves allow us to graph using a graphing calculator? Skills check, ability to: Accurately sketch parametric curves indicating starting point and direction Correctly use a graphing calculator to graph a parametric curve, setting appropriate window and t-step Eliminate the parameter for simple parametric functions Graph inverse relations Concept check: How are parametric functions related to rectangular functions? Identify real world examples where change should most accurately be described using a parameter. Grade 12 Honors Calculus, page 6 Topic 2.1 The Tangent & Velocity Problems Honors Calculus Grade 12 Unit 2: Limits & Derivatives Learning Objectives: Key Definitions, Skills and Concepts What is average rate of change? What is instantaneous rate of change? How are they different and how are they the same? Graphically, what corresponds to average and instantaneous rate of change? What are the units of rates of change? What is the difference quotient? Skills check, ability to: Correctly identify the average and instantaneous rates of change of a function that is expressed graphically and algebraically. This should include proper units. Concept check: Will the average rate of change and instantaneous rate of change of a function be the same? Why or why not? Identify the significance of a positive/negative rate of change. Grade 12 Honors Calculus, page 7 Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts Topic 2.2 The Limit of a Function What is a limit? What is appropriate limit notation? What is a one sided limit? Do all functions have limits? What would a graph look like where a limit either does not exist or is one sided? Skills check, ability to: Correctly identify the limit of a function given its graph Correctly identify the limit of a function given its equation Identify values at which the limit of a function does not exist Identify limits in a piecewise function Sketch a function given information about its limits Use a graphing calculator to evaluate limits of a function Find limits using direct substitution and factoring Concept check: What does it mean to say lim f ( x) L ? x c If lim f ( x) L , must f(c) = L? Why or why not? x c x2 4 and g(x) = x + 2, then we can say that the functions f and g are equal. x2 The statement: “Whether or not lim f ( x ) exists , depends on how f(a) is defined” is true: Sometimes, always True/False: If f(x) = xa or never Explain the difference between these two statements: f(a) = L and lim f ( x) L x a ( x 5)( x 5) ( x 5)( x 5) Explain why x 5 , but lim lim( x 5) x 5 x 5 ( x 5) ( x 5) True/False: As x increases to 100, f(x) = 1 gets closer to 0, so the limit as x goes to 100 of f(x) is zero. Be x prepared to justify your answer. If a function f is not defined at x = a, A) lim f ( x ) cannot exist xa B) lim f ( x ) could be 0 xa C) lim f ( x ) must approach xa Grade 12 Honors Calculus, page 8 Topic 2.4 Continuity Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What does it mean to say a function is continuous at a point, over an interval and at an endpoint? What are the types of discontinuities? What does the Intermediate Value Theorem say? Skills check, ability to: Identify where a function is continuous given its graph or its algebraic equation Sketch functions given information about intervals of continuity Find values for constants so that a piecewise function will be continuous at a given point Concept check: A drippy faucet adds one milliliter to the volume of water in a tub at precisely one second interval. Let f be the function that represents the volume of water in the tub at time t. Which of the following is true? (a) f is a continuous function at every time t (b) f is continuous for all t other than the precise instants when the water drips into the tub (c) f is not continuous at any time t (d) there is not enough information to discern anything about the continuity of f A drippy faucet adds one milliliter to the volume of water in a tub at precisely one second intervals. Let g be the function that represents the volume of the water in the tub as a function of the depth of the water, w, in the tub. Which of the following is true? (a) g is continuous at every depth w (b) there are some values of w at which g is not continuous (c) g is not continuous at any depth w (d) ) there is not enough information to discern anything about the continuity of g You decide to estimate e 2 by squaring the decimal approximation 2.71828… (a) This is a good idea because e is a rational number (b) This is a good idea because y = x 2 is a continuous function (c) This is a bad idea because e is irrational (d) This is a good idea because y = e x is a continuous function True/False: You were once three feet tall True/False: At some time since you were born, your weight in pounds equaled your height in inches True/False: Suppose that during the half time in a basketball game your team had 36 points. At some time during the first half, the team had to have exactly 25 points. Grade 12 Honors Calculus, page 9 Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts Topic 2.5 Limits involving Infinity What are infinite limits and limits at infinity? What are the definitions of vertical and horizontal asymptotes? Skills check, ability to: Identify infinite limits given the graph of a function or the equation of a function Identify limits at infinity given the graph of a function or the equation of a function Sketch the graph of a function given information about infinite limits and limits at infinity Find a formula for a function given horizontal and vertical asymptotes Concept Check: Why is lim sin x undefined? x What would the graph of f(x) look like if lim f ( x) ? x True/False: A graph can cross its horizontal asymptote. Grade 12 Honors Calculus, page 10 Topic 2.6 Tangents, Velocities and Other Rates of Change Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What is the tangent line drawn to a curve at a point? What is the secant line of a curve? What is the definition of the slope of a tangent line drawn to a curve at x=a and what is its relationship to the instantaneous rate of change of a function? What is the definition of velocity? Skills check, ability to: Find the slope of the tangent line drawn to a curve at a given point using the limit definition Find the equation of the tangent line drawn to a curve at a given point Find the velocity of a moving particle at a specific point given its position function Sketch a graph of the position of a moving particle given information about its velocity and a starting point Concept Check: Is the instantaneous rate of change of a function equal to the average of two average rates of change? Why or why not? 1 True/False: The function f(x) = x 3 is continuous at x=0 1 True/False: The function f(x) = x 3 is differentiable at x=0 The function y = f(x) measures the fish population in Blue Lake at time x, where x is measured in years since January 1, 1950. If f ′ (25) = 500, it means that (a) there are 500 fish in the lake in 1975 (b) there are 500 more fish in 1975 than there were in 1950 (c) on the average, the fish population increased by 500 fish per year over the first 25 years following 1950 (d) on January 1, 1975, the fish population was growing at a rate of 500 fish per year (e) none of the above Grade 12 Honors Calculus, page 11 Topic 2.6 Tangents, Velocities and Other Rates of Change (cont’d) Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts Let y = g(x) be a function that measures the water depth in a pool x minutes after the pool begins to fill. Then g ′ (25) represents: I) the rate at which the depth of the water in increasing 25 minutes after the pool starts to fill II) the average rate at which the depth changes over the first 25 minutes III) the slope of the graph of g at the point where x = 25 a) I only b) II only c) III only d) I and II e) I and III f) I, II and III The line tangent to the graph f(x) = x at (0,0) is a) y = 0 b) y = x c) does not exist d) is not unique. There are infinitely many tangent lines Grade 12 Honors Calculus, page 12 Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts Topic 2.7 Derivatives What is the derivative of a function at x=a? What is the geometric interpretation of the derivative of f at x=a? What is the interpretation of the derivative as a rate of change of a function? Skills check, ability to: Find f ′ (c) given f(x) as a graph or as an equation Correctly identify the units of f ′ (c) Use technology to find f ′ (c) Concept Check: Given common business applications including C(x), the cost of manufacturing x items, what is the proper interpretation of C ′ (c)? If f ′ (a) exists then lim f ( x ) xa Equals f(a) Equals f ′ (a) May not exist Explain the difference between these two statements: lim h 0 f ( x h) f ( x ) and ( x h) x f ( x h) f ( x ) ( x h) x Grade 12 Honors Calculus, page 13 Topic 2.7 Derivatives (cont’d) Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts Which of the following statements are always true? I) A function that is continuous at x = c must be differentiable at x=c II) A function that is differentiable at x = c must be continuous at x=c III) A function that is not continuous at x = c must not be differentiable at x = c IV) A function that is not differentiable at x = c must not be continuous at x = c a) None of them b) I and III c) II and IV d) I and IV e) II and III f) I, II, III and IV Let f be a function with f ′ (5) = 8. Which of the following statements must be true? a) f must be continuous at x = 5 b) f is definitely not continuous at x = 5 c) There is not enough information to determine if f is continuous at x = 5 Grade 12 Honors Calculus, page 14 Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts Topic 2.8 The Derivative As A Function What is the definition of f ′ (x)? How can we find f ′ (x) given f as a graph, a table or as an equation? What are the alternate notations for f ′ (x)? Are all functions differentiable everywhere? What is the relationship between being differentiable and continuous at a point? What is a second derivative and how is it found? Skills check, ability to: Given the graph of a function estimate f ′ (c) for various values of c Given the graph of a function, create a sketch of its derivative function Given an equation for f(x), find f ′ (x) algebraically Given a sketch of a function, identify where it is not differentiable and why Given a sketch of a function and its first and second derivatives, identify which is which Concept check: What is the derivative of a function telling us about the behavior of the function? True/False: The function f(x) = x has a derivative at x = 0 True/False: The function g(x) = x x has a derivative at x = 1 If f ′ (a) exists, lim f ( x ) xa i. it must exist, but there is not enough information to determine it exactly ii. equals f(a) iii. equals f ′ (a) iv. it may not exist Your mother says “If you eat your dinner, you can have dessert.” You know this means “If you don’t eat your dinner, you cannot have dessert.” Your calculus teacher says, “If f is differentiable at x, f is continuous at x.” You know this means a) if f is not continuous at x, f is not differentiable at x b) if f is not differentiable at x, f is not continuous at x c) knowing f is not continuous at x, does not give enough information to deduce anything about whether the derivative of f exists at x Grade 12 Honors Calculus, page 15 Topic 2.9 What Does f ′ Say About f? Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What is a local maximum and minimum? What are absolute extrema? What is concavity and what does it represent? What is an antiderivative? Skills check, ability to: Given the graph of a derivative function, f ′ , identify the places where f is increasing/decreasing, concave up/down, local max/mins, points of inflection Sketch a graph of a function given information about its derivative Sketch a graph of the antiderivative of a given function Concept check: What do you know about the value of the derivative function on those intervals where the original function f is increasing/decreasing? Explain why these observations make sense in terms of the slope of f When the graph of f ′ crosses the x-axis, what does this tell you about the graph of f? Explain, in terms of slope, why this happens. Based on the appearance of the graph, which one of these functions looks like it could be its own derivative? a) f(x) = sin x b) f(x) = cos x c) f(x) = e x -3 d) f(x) = e x5 e) none of the above Grade 12 Honors Calculus, page 16 Topic 3.1 Derivatives of Polynomials & Exponential Functions Honors Calculus Grade 12 Unit 3: Differentiation Rules Learning Objectives: Key Definitions, Skills and Concepts What are the derivatives of a constant function, a power function, a constant times a function, a sum/difference of functions and an exponential function? Skills check, ability to: Find derivatives of these types of functions Write the equation of tangents & normals Find higher order derivatives Concept check: Explain how differentiation and e are related. Use the derivative to draw conclusions about increasing/decreasing, concave up/down True/False: If f(x) = e 2 , then f ′ (x) = 2e True/False: An equation of the tangent line to the parabola y = x 2 at (-2,4) is y – 4 = 2x ( x + 2 ) Why is the natural exponential function y = e x used more often in calculus than the other exponential function y = a x ? 3.2 The Product & Quotient Rules What is the derivative of a function that is the product/quotient of two differentiable functions? Skills check, ability to: Find derivatives of this type Concept check: Understand when it is appropriate to use the product or quotient rule 3x 4 5 x 2 (i.e. f(x) = x 3 (x 2 +3x-4) and g(x) = 2x Be able to state the product/quotient rules in English Grade 12 Honors Calculus, page 17 Topic 3.3 Rates of Change in the Natural & Social Sciences Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What is the interpretation of the derivative in physics, chemistry, biology, economics, psychology, sociology etc? Skills check, ability to: Find the derivative given an initial function in one of these disciplines. Correctly state the units in which the derivatives should be expressed Understand the information that is conveyed by the numerical derivative at a given point Comment on the statement: “Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them” (Joseph Fourier) Concept check: If y = f(x) is a profit function measuring the amount of profit (in dollars) as a result of manufacturing and selling x basketballs, what is the significance of f ′ (550)? Make sure you use specific units. The water level, W(t), (where t is measured in hours) is falling at 3 inches every hour. Write an equation involving a derivative to describe this situation. 3.4 Derivatives of Trig Functions What are the derivatives of the trig functions? Skills check, ability to: Find the derivatives of trig functions as well as sum/difference and product/quotient functions involving trig functions Concept Check: Graphically demonstrate that the derivative of the sine function is the cosine function Using the derivative of the sine function determine increasing/decreasing. Compare these conclusions to the ‘unit circle’ definition of the sine function. Grade 12 Honors Calculus, page 18 Topic 3.5 The Chain Rule Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What is a composed function? When is the use of the chain rule appropriate? Write a complicated function as the composition of other simpler functions. What is the derivative of a composed function? Skills check, ability to: Correctly find the derivative of a composed function given algebraically, in a table, or graphically Concept Check: Be able to state the Chain Rule in English If f and g are both differentiable function a h = f A) f ′ (2) g ′ (2) B) f ′ (2) g ′(2) C) f ′ (g(2)) * g ′ (2) D) f ′ (g(x)) * g ′ (2) 3.6 Implicit Differentiation g, then h ′ (2) equals What is the difference between a function expressed explicitly and implicitly? When is it appropriate to use implicit differentiation? Skills check, ability to: dy Correctly find by implicit differentiation dx Concept Check: Grade 12 Honors Calculus, page 19 Topic 3.7 Derivatives of Logarithmic Functions Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What is a logarithmic function? How can it be expressed exponentially? What is logarithmic differentiation and when is its use appropriate? What is the definition of e as a limit? Skills check, ability to: Correctly find the derivative of a logarithmic function Correctly use logarithmic differentiation Concept check: Understand the difference between the derivative of a logarithmic function and logarithmic differentiation. Recognize that implicit differentiation is used to develop the formula for the derivative of a logarithmic function Why is the natural logarithmic function y = ln x used more often in calculus than the other logarithmic function y = log a x? 3.8 Linear Approximations What is the meaning of the phrase “approximating along a tangent line” and its connection to linear approximation. Discuss why one would choose to approximate a function when you can easily evaluate it on a calculator. Emphasize that this is the first, of many, applications of the derivative. Skills check, ability to: Linearize a function for a given value and use this to approximate the function for nearby values. Concept check: Some linearizations provide underestimates and some provide overestimates. What characteristic of the function will predict when you get each? The line tangent to the graph f(x) = sin x at (0,0) is y = x. This implies that A) sin (.005) .005 B) The line y = x touches the graph of f(x) = sin x at exactly one point (0,0) C) y = x is the best straight line approximation to the graph y = sin x for all x Grade 12 Honors Calculus, page 20 Topic 4.1 Related Rates Honors Calculus Grade 12 Unit 4: Applications of Differentiation Learning Objectives: Key Definitions, Skills and Concepts How can we calculate rates of change we cannot measure from rates of change that we already know? Skills check, ability to: Develop a mathematical model of a problem Write an equation relating the variable whose rate of change you seek with the variable whose rate of change is known Correctly differentiate both sides of an equation implicitly with respect to time Interpret a solution by translating the mathematical result into the problem setting to determine whether the result makes sense Concept check: Would it be reasonable to say that the rate at which water enters a drip coffeemaker is equivalent to the rate at which it leaves? What kind of coffee would that produce? What piece of a coffeemaker is designed so that there is no overflow? Gravel is poured into a canonical pile. The rate at which the gravel is added to the pile is: dV A) dt dr B) dt dV C) dr Peeling an orange changes its volume V. What does V represent? A) the volume of the rind B) the surface area of the orange C) the volume of the ‘edible’ part of the orange D) -1 times the volume of the rind Grade 12 Honors Calculus, page 21 Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts Topic 4.1 Related Rates (cont’d) It is a beautiful spring evening. You are reading a calculus book and you arrive at the following: A ladder 10 feet long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall? Your enthused roommates don’t read the rest of the example and proceed as follows: wall y x “We want to find ground dy dy . So we set up x 2 + y 2 = 100. Now we want when dt dt dx = 1 and x = 6. Substituting x = 6 gives us 36 + y 2 = 100 or y 2 = 64. Now we take derivatives to get dt dy dy 2y = 0, giving = 0. This doesn’t make any sense. What went wrong? dt dt Grade 12 Honors Calculus, page 22 Topic 4.2 Maximum and Minimum Values Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What are absolute extreme values of a function? What are local extreme values of a function? What is a critical point of a function? Under what circumstances does a function NOT have absolute extrema? What is the Extreme Value Theorem? Skills check, ability to: Given the equation of a function, find local and absolute extreme values using transformations and graphs Sketch functions given information about extreme values Find absolute max/min of a function defined on a closed interval Concept check: Is it always true that f ′ (c) = 0 if f has a local extreme value at c? If a function is defined on a closed interval, explain why it must have a global maximum? True/False: Every global max/min is a local max/min True/False: If f(x) is continuous on a closed interval, then it is enough to look at the points where f ′ (x) = 0 in order to find its absolute maximum and minimum. Be prepared to justify your answer. If f is continuous on [a,b], then A) there must be number m and M such that m f ( x) M for all x in [a,b] B) there must be local extreme vales, but there may or may not be an absolute maximum or an absolute minimum value for the function C) any absolute max or min would be at either endpoint of the interval, or at places in the domain where f ′ (x) = 0 4.3 Derivatives & Shapes of Curves What is the Mean Value Theorem? What is the relationship between the first derivative of a function and whether it is increasing or decreasing? What is the first derivative test? What is the definition of concavity? What are the tests that determine the concavity of a function? Skills check, ability to: Given the graph of a function, indicate intervals of increasing/decreasing and concave up/down Given the graph of the derivative of a function, indicate intervals at which the function is increasing/decreasing and concave up/down Given the equation of a function, find local/global max/min, intervals of increasing/decreasing, intervals of concave up/down and a sketch using the derivative Grade 12 Honors Calculus, page 23 Topic 4.3 Derivatives & Shapes of Curves (cont’d) Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts Concept check: What result tells us that if a biker travels 45 km in 3 hours, then her speedometer must have read 15 km/hr at least once during the trip? Can we guarantee that it will read 12 km/hr during the trip? How about 18 km/hr? Is it true that if f ′ (c) = 0, x = c is a maximum? Is it true that if f ′′ (c) = 0, x = c is a point of inflection. What can you conclude about x =c if f ′ (c) does not exist? True/False: If f ′ (c) = 0 and f(c) is not a local maximum, it must be a local minimum. A continuous function f has domain [1, 25] and range [3, 30]. If f ′ (x) < 0, for all x between 1 and 25, what is f (25)? Suppose f is continuous on [0, 6] and satisfies the following: x 0 3 5 6 f -1 4 -1 -3 f′ 5 0 -8 0 f ′′ -1 -3 DNE 3 AND x 0<x<3 3<x<5 5<x<6 f′ + f′′ + Identify all points of inflection The following is known about the function f: on the interval [-5, 5], f is continuous and differentiable and f ′ (x) = (x+1)(2x+1)(x+3) 2 . Briefly explain each of the following conclusions: 1 A) There is a local minimum of f at x = 2 B) There is a horizontal tangent but no extrema at x = -3 C) If f(2) = 7, then f(3) > 7 Water is being poured into a “Dixie Cup” (a standard cup that is smaller on the bottom than at the top). The height of the water in the cup is a function of the volume of the water in the cup. The graph of this function is A) increasing concave up B) increasing concave down C) a straight line with positive slope Grade 12 Honors Calculus, page 24 Topic 4.4 Graphing with Calculus and Calculators NOTE: Although this is optional material, it does remind students that there are instances when a calculator should not be trusted. Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts Why would anyone use Calculus to sketch a curve when it can be done accurately and quicker using a graphing calculator? Skills check, ability to: Combine graphical methods and analytic methods of calculus Distinguish between the graphing calculator as a means of estimating local extremes and inflections points, contrasted with the use of calculus for precise computation of such points Concept check: Is the calculator always reliable? Of what use can calculus be when you cannot find a ‘good’ viewing window for a function? Grade 12 Honors Calculus, page 25 Topic 4.6 Optimization Problems Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What does it mean to optimize a function? What kinds of functions would one most likely want to minimize/maximize? Skills check, ability to: Develop a mathematical model of a situation Clearly identify the quantity to be maximized/minimized Write a function whose extreme value gives the information sought Graph this function Using calculus methods, identify the critical points and endpoints of this function Interpret your solutions into the problem setting and decide whether your results make sense Concept check: The steps of the optimization process are listed in random order. Number them from 1 to 8 as they should be completed: Justify that your solution provides a max or min Find the derivative of the varying quantity If necessary, substitute from the fixed quantity into the varying quantity Make sure you have answered the original question, with appropriate units of measure Define variables to be used Read and understand the problem, noting especially what is to be optimized Find the zeros of the derivative equation Write equations for all fixed and varying quantities 4.7 Applications to Business & Economics NOTE: Although this is optional material with respect to the AP Exam, it should not be omitted in this class as many of our students will be pursuing business studies. What is a cost function, a marginal cost function, an average cost function, a demand function, a revenue function, a marginal revenue function, a profit function and a marginal profit function? What are the geometric and graphical interpretations of average and marginal cost? Skills check, ability to: Given a cost and revenue sketch, identify the point at which the profit is maximized and sketch a profit function Create a linear demand function given information about cost and sales Use calculus to find the level of production that will maximize profits Grade 12 Honors Calculus, page 26 Topic 4.7 Applications to Business & Economics (cont’d) Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts Concept check: Here are three different cost functions. Suggest reasons why they look the way they do: Here are three different revenue functions. Suggest reasons why they look the way they do: Here is a sketch of typical revenue and cost functions. Find the maximum profit graphically (where the difference between the curves is greatest and revenue is larger than cost). Show that this maximum profit always occurs at the point where the tangent lines to the curves are parallel to each other. Then sketch the profit function and the marginal profit function. Grade 12 Honors Calculus, page 27 Topic Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts 4.8 Newton’s Method How does a graphing calculator find the real zeros of a function? What is meant by an iterative process? NOTE: Although this material is optional for the AP Exam, it is recommended here as it explains how a calculator finds zeros. Skills check, ability to: Write an equation whose root is the value you seek Choose a reasonable initial estimate Either by hand or by programming a graphing calculator find successive approximations until the desired level of accuracy is reached Concept check: Under what circumstances would Newton’s Method fail? Suppose your first guess in using Newton’s Method is lucky in the sense that x 1 is a root of f(x) = 0. What happens to x 2 and later approximations? Newton’s Method is a cool technique, because A) it can help us get decimal representations of number like B) it can be used to find a solution to x 7 3x3 1 C) both (A) and (B) Grade 12 Honors Calculus, page 28 4 3, 8 5 , and 5 13 Topic Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts 4.8 Newton’s Method (cont’d) Grade 12 Honors Calculus, page 29 Topic 4.9 Antiderivatives Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What is an antiderivative? Does every function have a unique antiderivative? What is a slope field and how is it used to find an antiderivative? Skills check, ability to: Find general antiderivatives Find antiderivatives given a function and an initial condition Given a slope field, use it to determine an antiderivative if you know an initial condition Create a slope field given an equation for f ′ (x) Given a velocity function, find the position function Concept check: If F (x) is an antiderivative of f (x), what is F ′ (0) ? True/False: An antiderivative of a product of functions, fg, is an antiderivative of f times an antiderivative of g True/False: An antiderivative of the sum of two functions, f + g, is the antiderivative of f plus the antiderivative of g If f is an antiderivative of g and g is an antiderivative of h then, a. h is an antiderivative of f b. h is the second derivative of f c. h is the derivative of f ′′ Grade 12 Honors Calculus, page 30 Honors Calculus Grade 12 Unit 5: Integrals Topic Learning Objectives: Key Definitions, Skills and Concepts 5.1 Area And Distances What are the ways to find the area of irregularly shaped regions? How can one find the area under the graph of a nonnegative continuous function? What is the definition of the area of a region under the graph of a continuous function? Skills check, ability to: -use a given number of rectangles to calculate the area under the graph of a nonnegative continuous function -distinguish between right and left hand approximations, midpoint approximations and trapezoidal approximations -use a graphing calculator to approximate area under the graph of a nonnegative continuous function using a large number of rectangles -Express the area under the graph of a nonnegative continuous function as a limit -Correctly write sums using Sigma Notation Concept check: -True/False: If a piece of string has been chopped into n small pieces and the i th piece is xi inches long, then the total n length of the string is exactly x i 1 i -Demonstrate using a constant velocity that the area under the curve represents the total distance traveled -Explain why this would generalize to a velocity that was not constant -You want to estimate the area underneath the graph of a positive function by using four rectangles of equal width. The rectangles that must give the best estimate of this area are those with heights from the A) Left Endpoints B) Midpoints C) Right Endpoints D) Not enough information Grade 12 Honors Calculus, page 31 Topic 5.2 Definite Integral Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What is a definite integral? What are its properties? How can the area under a curve be expressed as a definite integral and the limit of Riemann sums? How can the area under a curve be computed using numerical integration processes? Skills check, ability to: -Evaluate Riemann sums for a small number of subintervals -Understand what the notation b a f ( x)dx is referring to -Evaluate definite integrals by interpreting them in terms of areas (this can be for functions that are positive and negative) -Evaluate definite integrals based on their properties (i.e. -Simplify definite integrals based on their properties (i.e. 4 2 7 3 1 10 f ( x)dx f ( x)dx f ( x)dx ) 2 4 f ( x)dx 15 and 7 4 f ( x)dx 5 , find 4 3 f ( x)dx ) -Be able to use a graphing calculator to evaluate definite integrals Concept check: -Why is it reasonable that b a a f ( x)dx f ( x)dx ? b Based upon areas, explain why if f(x) 0 for a x b, then -If b a b a f ( x)dx must be positive f ( x)dx 0 , must f(x) be positive for all values of x in [a,b]? -Give a convincing argument to show that 2 0 (3 sin x)dx 6 -Which of the following quantities would NOT be represented by the definite integral 8 0 70dt ? a) The distance traveled by a train moving at 70 mph for 8 minutes b) The volume of ice cream produced by a machine making 70 gallons of ice cream per hours for 8hours c) The length of track traveled by a snail traveling at 70 cm per hour for 8 hours d) The total sales of a company selling $70 of merchandise per hour for 8 hours e) The amount the tide has risen 8 minutes after low tide if it rises at the rate of 70 mm per minute during that period -True or False: If b a f ( x)dx = 0 then f(a) = f(b) Grade 12 Honors Calculus, page 32 Topic 5.3 Evaluating Definite Integrals Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What is the evaluation Theorem? What is the difference between a definite and indefinite integral? What is the Net Change Theorem? What unit is used for b a f ( x)dx ? Skills check, ability to: Evaluate definite integrals by using the Evaluation Theorem Find general indefinite integrals Concept check: If you know that the population of an ant colony starts with 100 ants and increases at the rate of P ′ (t) what does 100 + 4 0 P '(t )dt represent? What does b a P '(t )dt represent? True/False: if f is continuous on the interval [a,b], then b a f ( x)dx is a number Read the following four statements and choose the correct answer below: If f is continuous on the interval [a,b] then: b f ( x)dx is the area bounded by the graph of f, the x-axis and the lines x = a and x = b ii) f ( x)dx is a number iii) f ( x)dx is an antiderivative of f(x) iv) f ( x)dx may not exist i) a b a b a b a A) B) C) D) (ii) only (i) and (ii) only (i) and (iii) only (iv) only Water is pouring out of a pipe at the rate of f(t) gallons/minute. You collect the water that flows from the pipe between t=2 and t=4. The amount of water you collect can be represented by: A) 4 2 f ( x)dx B) f(4) –f(2) C) (4-2)f(4) D) the average of f(4) and f(2) times the amount of time that elapsed Grade 12 Honors Calculus, page 33 Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts Topic 5.3 Evaluating Definite Integrals (cont’d) A sprinter practices by running various distances back and forth in a straight line in a gym. Her velocity at t seconds is given by the function v(t). What does 60 0 v(t ) dt represent? A) the total distance the sprinter ran in one minute B) the sprinters average velocity on one minute C) the sprinter’s distance from the starting point after one minute D) None of the above What is the difference between total distance traveled and net displacement? 5.4 The Fundamental Theorem of Calculus What are the two branches of calculus and what problems gave rise to each? What is The Fundamental Theorem of Calculus? What is the derivative of a function defined as an integral? Skills check, ability to: Given the graph of a piecewise function f(t), and the fact that g(x) = x 0 f (t )dt , evaluate g(x) for specific values of x and sketch a rough graph of g identifying intervals on which g is increasing/decreasing and where local max/min values occur Find the derivative of a function defined as an integral Grade 12 Honors Calculus, page 34 Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts Topic 5.4 The Fundamental Theorem of Calculus (cont’d) Below is a graph of a function f Let g(x) = A) B) C) D) x 0 f (t )dt . Then for 0 < x < 2, g(x) is increasing concave up increasing concave down decreasing concave up decreasing concave down Using the same graph as above, and the same definition of g. Then: A) g(0) = 0, g ′ (0) = 0 and g ′ (2) = 0 B) g(0) = 0, g ′ (0) = 4 and g ′ (2) = 0 C) g(0) = 1, g ′ (0) = 0 and g ′ (2) = 1 D) g(0) = 0, g ′ (0) = 0 and g ′ (2) = 1 Grade 12 Honors Calculus, page 35 Topic 5.4 The Fundamental Theorem of Calculus (cont’d) Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts Concept check: Explain what is meant by the statement that “differentiation and integration are inverse processes” The graph of the derivative F′ of some function F is given. If you are told that F(20) = 150, estimate the maximum value attained by F Grade 12 Honors Calculus, page 36 Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts Topic 5.5 The Substitution Rule How do you find an antiderivative of a function that is the product of two functions? What is the substitution rule for definite integrals? Skills check, ability to: Evaluate integrals given the appropriate substitutions Evaluate integrals without being given the appropriate substitution Concept check: What is the differentiation rule that helps us understand why the Substitution rule works? True/False: The following can be found using the substitution rule: 2 A) xe x dx B) e x dx 2 C) 1 4 x ( x 2 4)dx Given: A) 1 2 0 6 0 sin 2 x cos 2 xdx . Letting u = sin 2x, the integral becomes: udu 1 12 B) udu 2 0 C) 2 3 2 0 3 2 udu D) E) 1 23 udu 2 0 0 udu Grade 12 Honors Calculus, page 37 Topic 5.6 Integration By Parts(Optional) Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts What is integration by parts used for? What is the formula for integration by parts? What is the tabular method for integration by parts? Skills check, ability to: Evaluate an integral given expressions for u and dv Evaluate an integral without being given expressions for u and dv Concept check: True/False: To evaluate udv the primary criterion is that dv must be something you can integrate. An additional criterion is that u should be something that gets simpler when it is differentiated What rule of differentiation helps us to understand why integration by parts works? Grade 12 Honors Calculus, page 38 Honors Calculus Grade 12 Unit 6: Applications of Integration Topic 6.1 More About Areas Learning Objectives: Key Definitions, Skills and Concepts How is the area between two curves found? When is it appropriate to integrate with respect to y as opposed to x? Skills check, ability to: Sketch the region enclosed by two curves Decide whether to integrate with respect to x or y Find the area of the region Concept check: Is the following statement always, sometimes or never true? The area of the region between the graphs of the continuous functions y = f(x) and y = g(x) and the vertical lines x = a and x = b (a < b) is: b [ f ( x) g ( x)]dx a Grade 12 Honors Calculus, page 39 Topic 6.2 Volumes Honors Calculus Grade 12 Learning Objectives: Key Definitions, Skills and Concepts How do we find the volume of a solid generated by rotating the region bounded by given curves about a specified line? Skills check, ability to: Correctly calculate a volume of a solid generated by rotating a region bounded by given curves about a specified line Concept check: Here are the steps, in random order, involved in finding volume by the method of slicing. Put the steps in the appropriate order. A) Find the limits of integration B) Find a formula for A(x) C) Sketch the solid and a typical cross section D) Integrate A(x) to find the volume What kind of regions, when rotated about a specified line, will result in volumes that can be computed using the disk versus washer method? True/False: The volume of the solid generated by revolving the region bounded by y = x and the lines y = 2 and x = 0 about the x-axis is equal to the volume of the solid generated by revolving the region bounded by y = x and the lines y = 2 and x = 0 about the y-axis. True/False: The region enclosed by the y-axis, the line y = 2, and the curve y = axis. The volume of the solid is given by 2 0 y 2 dy Grade 12 Honors Calculus, page 40 x is revolved about the y- Honors Calculus Grade 12 Recommended Unit Sequencing and Pacing Guide Timeframe Q1 Unit 1: Functions and Models 1.1 Four Ways to Represent A Function 1.2 Mathematical Models: A Catalog of Essential Functions 1.3 New Functions From Old Functions 1.4 Graphing Calculators and Computers 1.5 Exponential Functions 1.6 Inverse Functions and Logarithms 1.7 Parametric Curves Unit 2: Limits & Derivatives 2.1 The Tangent & Velocity Problems 2.2 The Limit of a Function 2.3 Calculating Limits Using Limit Laws Q2 Unit 2: Limits & Derivatives 2.4 Continuity 2.5 Limits Involving Infinity 2.6 Tangents, Velocity and Other Rates of Change 2.7 Derivatives 2.8 The Derivative As A Function 2.9 What Does f ′ Say About f? Unit 3: Differentiation Rules 3.1 Derivatives of Polynomial & Exponential Functions 3.2 The Product & Quotient Rules Midterm Grade 12 Honors Calculus, page 41 Honors Calculus Grade 12 Timeframe Q3 Unit 3:Differentiation Rules 3.3 Rates of Change in the Natural & Social Sciences 3.4 Derivatives of Trig Functions 3.5 The Chain Rule 3.6 Implicit Differentiation 3.7 Derivatives of Logarithmic Functions 3.8 Linear Approximations Unit 4: Applications of Differentiation 4.1 Related Rates 4.2 Maximum & Minimum Values 4.3 Derivatives & The Shape of Curves Q4 Unit 4: Applications of Differentiation 4.4 Graphing With Calculus & Calculators 4.6 Optimization Problems 4.7 Applications to Business & Economics 4.8 Newton’s Method 4.9 Antiderivatives Unit 5: Integrals 5.1 Area & Distances 5.2 The Definite Integral 5.3 Evaluating Definite Integrals 5.4 The Fundamental Theorem of Calculus 5.5 The substitution Rule 5.6 Integration By Parts Unit 6: Applications of Integration 6.1 More About Areas 6.2 Volumes Final Grade 12 Honors Calculus, page 42