Download Calculus v. 2016

2015 - 2016
Calculus
This course is designed to provide a firm background and understanding of the basic concepts of calculus;
including limits, differentiation, applications of derivatives, exponential/logarithmic functions, integration,
applications of integration and trigonometric functions.
Course Information:
Frequency & Duration: Daily for 42 minutes
Text: Barnett, Raymond, et al. Calculus: For Business, Economics, Life Sciences, and Social
Sciences. 9th ed. Upper Saddle River: Prentice Hall, 2002
Calculus
v. 2015 - 2016
Content: Functions
Duration: Aug. /Sept (2 weeks)
Essential What information needs to be known / found in order to graph a function?
Question: What strategies can be used to solve for unknowns in algebraic equations?


Skill:

Graph all elementary functions and transformations of these functions.
Summarize the key components of a parent, polynomial, rational, exponential, and logarithmic
function’s graph (including intercepts, asymptotes, end behavior, and turning points/extrema).
Graph a quadratic (standard form or vertex form), rational, polynomial, exponential, and logarithmic
function using the key components and/or transformations.
Solve a polynomial equation by factoring (including perfect square, difference of squares, sum of
cubes and difference of cubes), quadratic formula and factoring by grouping.
Solve a rational, radical, exponential, and logarithmic equation.




Graph 𝑦 = −2 √𝑥 + 3 − 1
Graph 𝑦 = 𝑒 −𝑥 + 3
Graph 𝑦 = ln(𝑥 + 3) − 4
−3𝑥
Analyze and graph 𝑓(𝑥) = (𝑥+3)2


Assessment: 
3



Analyze and graph 𝑓(𝑥) = 2𝑥 4 − 5𝑥 3 − 4𝑥 2 + 3𝑥 + 6
Solve: 𝑥 2 − 5𝑥 = 6
Solve: 5(3𝑥 − 7)3 + 𝑥(3𝑥 − 7)2 = 0
2
1
Solve: = 5 + 2

Solve: 3 + √3𝑥 + 1 = 𝑥
𝑥
𝑥 +𝑥
Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Chapter 1
Resources: & 2 (pages 3-125, 596-647)
Standards: This content is beyond the scope of the PA Core Mathematical Standards.
Vocabulary:
Comments:
Exponential Function – Any function in the form: 𝑓(𝑥) = 𝑏 𝑥 , where 𝑏 is any real number such that
𝑏 > 0 aand 𝑏 ≠ 1; Logarithmic Function – Any function in the form: 𝑓(𝑥) = log 𝑏 𝑥, where 𝑏 is any
real number such that 𝑏 > 0 aand 𝑏 ≠ 1; Polynomial Function– An function in the form 𝑓(𝑥) =
𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 … 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0 . Where 𝑛 is an integer and 𝑎 is a number; Quadratic
Function– A function in the form 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 or 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘; Radical
Function– A function that contains a radical expression in the radicand; Rational Function – An
algebraic fraction such that the numerator and denominator are polynomials
Calculus
v. 2015 - 2016
Content: Evaluating Limits
Duration: Sept./Oct. (4 Weeks)
is the limit of a function and how can the limit of a function be computed? (What is the
Essential What
best/proper method in order to determine this?)
Question: How can limits be used to explain the asymptotes and continuity of a function?



Skill: 







Assessment:
Resources:




Define and understand the limit of a function.
Determine the one sided limits of a function.
Evaluate the limit of a function at any point on the graph of a function (Graphically, tables, and
algebraically).
Determine when a limit does and does not exist.
Evaluate infinite limits and limits at infinity.
Construct a possible graph for a function given different limits.
Understand the definition of continuity in terms of limits
Define limit.
Use a graph to determine the value of a limit.
Make a table of values to determine the value of the limit.
Evaluate a limit using algebraic techniques (simplification, cancellation, rationalization and
multiplying by conjugates).
Determine the limit at a point of discontinuity (holes, vertical asymptotes and breaks).
Determine the end behavior of the graph (limit at infinity)
Given the limit values of a function create a possible graph for the function.
𝑥−3
Where is the function, (𝑥) =
, discontinuous? Use the limit definition to explain why.
5−𝑥
Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Section 32 (pages 144-161)
Calculus: AP Edition. Pearson: Chapter 2 (pages 61-119)
Graphing Calculator
Standards: This content is beyond the scope of the PA Core Mathematical Standards.
Continuity– A function is continuous at a point if the following three conditions are met: 1. lim 𝑓(𝑥)
𝑥→𝑎
exists; 2. 𝑓(𝑎) exists; 3. lim 𝑓(𝑥) = 𝑓(𝑎); Indeterminate Form– When direct substituting to evaluate
0
Vocabulary:
Comments:
𝑥→𝑎
a limit the result is . This means that the limit cannot be determined with the function in this form.
0
More must be done to evaluate the limit; Infinite Limits– A limit that is equal to ±∞. The function
grows without bound as x approaches a given number; Limit– The value that y “approaches” as x
approaches a given number; Limits at Infinity– The value that y “approaches” as x approaches ±∞;
One-Sided Limit– The value that y “approaches” as x approaches a given number form one side of the
x-value
Calculus
v. 2015 - 2016
Content: Rates of Change/Derivative
Definition
Duration: October (3 Weeks)
Essential What is the difference between Average Rate of Change and Instantaneous Rate of Change?
Question: How do you calculate the slope of a non-linear function?
Skill:











Assessment:


Graphically interpret the average rate of change as the slope of a secant line.
Determine the average rate of change.
Graphically interpret the instantaneous rate of change as the slope of a tangent line or slope of the
graph.
Determine the instantaneous rate of change graphically and algebraically using the four step process.
Define a derivative and know the formula for a derivative.
Calculate a derivative using the definition and the four step process.
Given a function and two x-values calculate the average rate of change between the two points.
Relate the slope of a secant line to the difference quotient formula.
Given a function, calculate the instantaneous rate of change at a given x-value.
Estimate the instantaneous rate of change from a graph.
Know and explain the formula for finding the instantaneous rate of change.
o 𝑓(𝑥) = 2𝑥 + 3 at 𝑥 = −4
o 𝑡(𝑥) = 3 − 7√𝑥 at 𝑥 = −1
Know and explain the formula for a derivative.
Given a function calculate the derivative and find the slope at a given point.
o Find 𝑦′ given 𝑦 = 1 − 𝑥 2
1
o Find 𝑘 ′ (𝑥) given 𝑘(𝑥) =
2−𝑥
o Find 𝑔′ (𝑥) given 𝑔(𝑥) = 24
Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Section 3-
Resources: 2 (pages 131-144 & 162-175)
Calculus: AP Edition. Pearson: Chapter 2 (pages 136-153)
Standards: This content is beyond the scope of the PA Core Mathematical Standards.
Average Rate of Change– Slope of a secant line (𝑚 =
𝑓(𝑎)−𝑓(𝑏)
𝑎−𝑏
); Derivative– A function that gives
𝑓(𝑥+ℎ)−𝑓(𝑥)
); Difference
Vocabulary: the slope of a tangent line at any point in the domain of the function (lim
ℎ
ℎ→0
Quotient– (
Comments:
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
); Instantaneous Rate of Change– Slope of a tangent line (lim
ℎ→0
𝑓(𝑎+ℎ)−𝑓(𝑎)
ℎ
)
Calculus
v. 2015 - 2016
Content: Rules for Differentiation
Duration: November (4 Weeks)
Essential How do you find the slope of the tangent line at any point on the graph of the function?
Question:

Skill:



𝑑𝑥

Determine when a function is non-differentiable (graphically and algebraically).
Write the equation of the tangent line at a point on the graph of a function.
Determine where a function has a horizontal tangent line (graphically and algebraically).

Find the derivative of 𝑓(𝑥) = (





Find the derivative of 𝑓(𝑥) = (ln 𝑥 + 𝑒 −𝑥 )
Find the derivative of 𝑦 = log(2 − 𝑥)
2
Find the derivative of 𝑓(𝑥) = 32𝑥+𝑥
Find the derivative of 𝑦 = √ln(𝑥 2 − 2𝑥)
𝑒𝑥+5
Find the derivative of 𝑓(𝑥) =
2

Find

Given a function 𝑓(𝑥), find the 𝑥 values for which the function is non-differentiable and explain
why the derivative does not exist at these points.
Given a function and an 𝑥 value find the equation of a tangent line at that 𝑥 value.

Assessment:
Know and understand the rules for differentiation. Including constant, constant times a function,
power, sum and difference, product, quotient, chain, logarithmic and exponential rules.
Evaluate the derivative of a function using the rules for differentiation.
𝑑𝑦
Use implicit differentiation to calculate .

𝑑𝑦
𝑑𝑥
3
2𝑥
𝑥
+ )
3
ln 𝑥
if 𝑥 = 𝑒 𝑦 + 7
2

 Write the equation of the tangent line at 𝑥 = 8 for the function 𝑓(𝑥) = 3𝑥 3 − 5 √𝑥
 Write the equation of the tangent line at 𝑥 = 𝑒 for the function 𝑓(𝑥) = ln 𝑥 2
Given a function, algebraically find where it has a horizontal tangent line. (𝑓 ′ (𝑥) = 0)
𝑥
 𝑓(𝑥) = (𝑥+3)4


Resources:
𝑔(𝑥) =
3
𝑥
ln 𝑥
Given the graph of a function, determine where 𝑓 ′ (𝑥) = 0 and where 𝑓 ′ (𝑥) does not exist.
Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Sections
3.4, 3.5, 3.6, 5.2, 5.3 & 5.4 (pages 175 – 204 & 322 – 343 & 345 – 351)
Calculus: AP Edition. Pearson: Sections 3.3, 3.4 & 3.7- 3.9 (pages 153 – 221)
Standards: This content is beyond the scope of the PA Core Mathematical Standards.
Calculus
Vocabulary:
Comments:
v. 2015 - 2016
Differentiable– The derivative of the function exists at a given point(s); Differentiate– Find the
derivative of the function; Differentiation– The process of finding a derivative; Explicit Equation– An
equation that defines the dependent variable (𝑦) in terms of the independent variable(𝑥);Implicit
Equation– An equation that does not define the dependent variable (𝑦) in terms of the independent
variable (𝑥); Non-differentiable– The derivative of the function does not exist at a given point(s). A
function is non-differentiable at any point where the graph is discontinuous, has a sharp point, has a
vertical tangent line or at the end points.
Calculus
v. 2015 - 2016
Content: Derivative Applications
Duration: December (3 Weeks)
Essential How can a derivative be applied to a real world situation (business, economics, science, etc)?
Question:
Skill:
Assessment:




Differentiate an equation with respect to time (𝑡).
Use differentiation to solve related rate problems.
Relate distance, velocity, and acceleration using differentiation.
Use derivatives to analyze business equations.

Given a function, 𝑥 value, and

A construction worker pulls a 16-foot plank up the side of a building under construction by means
of a rope tied to the end of the plank. If the worker pulls the rope at a rate of 0.5 ft/sec. How fast
is the end of the plank sliding along the ground when it is 8 feet from the wall?
Given the distance equation, find the velocity equation and acceleration equation.
Interpret the slope of the distance and velocity function.
Given the price and cost equations, find the revenue and profit equations.
Find the marginal cost, marginal revenue, marginal profit, average cost, average revenue, average
profit, marginal average cost, marginal average revenue and marginal average profit, given the price
and cost equation.
Interpret the result from evaluating the derivative of a business function.
 Find 𝑅′ (500) and interpret.
 Find 𝐶̅ ′(1000) and interpret.
 Find 𝑃̅ (200)and interpret.





Resources:
𝑑𝑥
𝑑𝑡
. Find
𝑑𝑦
𝑑𝑡
Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Sections
3.4, 3.7 & 5.5, (pages 181 – 188 & 352 – 360)
Calculus: AP Edition. Pearson: Sections 3.6, 3.11 (pages 181 – 191 & 232 – 240)
ACTIVITY: Tootsie Pop Lab from A Watched Cup Never Cools from Key Curriculum Press.
Standards: This content is beyond the scope of the PA Core Mathematical Standards.
Calculus
Vocabulary:
Comments:
v. 2015 - 2016
Average (Cost, Revenue, Profit)– The average cost, revenue or profit per item; Marginal Average
(Cost, Revenue, Profit)– The rate at which the average cost, average revenue or average profit is
changing; Marginal (Cost, Revenue, Profit)– The rate at which the cost, revenue or profit is changing.
Or, the approximate cost, revenue, or profit of the next item
Calculus
v. 2015 - 2016
Content: Graphical Derivative Applications
Duration: January (4 Weeks)
Essential How can the concept of a derivative be used to interpret/explain the graph of a function?
Question:
Skill:




Determine where a function is increasing or decreasing.
Find the local extrema occur of a function using the first and second derivative test.
Determine the concavity and inflection points of a function.
Draw an accurate graph of a function and its derivative.

Use the relationship between 𝑓(𝑥) and 𝑓 ′ (𝑥) to answer graphical questions given 𝑓(𝑥), and 𝑓 ′ (𝑥).
o Given a graph of 𝑓(𝑥) determine where 𝑓′(𝑥) is negative.
o Given a graph of 𝑓 ′ (𝑥) determine where 𝑓(𝑥) increasing.
o Given a graph of 𝑓 ′ (𝑥) determine where 𝑓(𝑥) has a local minimum.
Use the relationship between 𝑓(𝑥), 𝑓 ′ (𝑥), and 𝑓 ′′ (𝑥) to answer graphical questions given 𝑓(𝑥),
𝑓 ′ (𝑥), and 𝑓 ′′ (𝑥).
o Given a graph of 𝑓(𝑥) determine where 𝑓′(𝑥) is decreasing.
o Given a graph of 𝑓 ′ (𝑥) determine where 𝑓 ′′ (𝑥) positive.
o Given a graph of 𝑓 ′′ (𝑥)determine where does 𝑓(𝑥) have inflection points.
Given an equation for 𝑓(𝑥) determine where the function is increasing and decreasing by making a
sign chart for 𝑓 ′ (𝑥).
3
o 𝑓(𝑥) = 2
𝑥 −9
o 𝑓(𝑥) = 𝑥𝑒 −𝑥
Given an equation for 𝑓(𝑥) use the first derivative test to interpret the sign for 𝑓 ′ (𝑥) and determine
where the function has local extrema.



Assessment:
o
o





2
5
𝑓(𝑥) = 𝑥 3 − 𝑥 3
ln 𝑥
𝑓(𝑥) =
, 𝑥>0
𝑥
Given an equation for 𝑓(𝑥) determine where the function is concave up, concave down and where
inflection points occur by making a sign chart for 𝑓 ′′ (𝑥).
3
o 𝑓(𝑥) = √(𝑥 − 3) + 1
o 𝑓(𝑥) = ln(𝑥 2 − 2𝑥 + 10)
Given an equation for 𝑓(𝑥) use the second derivative test to determine where the function has local
extrema.
o 𝑓(𝑥) = (𝑥 − 2)(3 − 𝑥)5
9
o 𝑓(𝑥) = 4 + 𝑥 +
𝑥
Given a sign chart for 𝑓 ′ (𝑥) and 𝑓 ′′ (𝑥), construct an accurate graph of 𝑓(𝑥) and 𝑓 ′ (𝑥).
Given the graph of 𝑓 ′ (𝑥), construct a sign chart for 𝑓 ′ (𝑥) and 𝑓 ′′ (𝑥) then construct an accurate
graph for 𝑓(𝑥).
Given the graph of 𝑓(𝑥), construct a sign chart for 𝑓 ′ (𝑥) and 𝑓 ′′ (𝑥) the construct an accurate
graph for 𝑓 ′ (𝑥).
Calculus
v. 2015 - 2016
Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Sections
Resources: 4.2 & 4.3, (pages 242 – 274)
Calculus: AP Edition. Pearson: Sections 4.1 & 4.2 (pages 246 – 271)
Standards: This content is beyond the scope of the PA Core Mathematical Standards.

Vocabulary:
Comments:
Concave up– When the graph of a function opens up. This occurs when 𝑓 ′′ (𝑥) is positive or when
𝑓 ′ (𝑥) is increasing; Concave down– When the graph of a function opens down. This occurs when
𝑓 ′′ (𝑥) is negative or when 𝑓 ′ (𝑥) is decreasing; Critical values– Partition numbers for 𝑓 ′ (𝑥) that are in
the domain of 𝑓(𝑥); Decreasing– When the slope (or derivative) of a function is negative; First
derivative test– A method to find the local extrema of a function by interpreting the sign chart for
𝑓 ′ (𝑥); Increasing– When the slope (or derivative) of a function is positive; Inflection point– A point
on the graph of a function where the concavity changes. At this point, 𝑓 ′′ (𝑥) = 0 and 𝑓 ′ (𝑥) has local
extrema; Local extrema– Where the graph of a function has a local minimum or local maximum. These
occur when the derivative changes form positive to negative or vice versa; Partition numbers
(𝒇𝒐𝒓 𝒇′ (𝒙))– Points where the function (𝑓(𝑥)) is equal to zero or discontinuous; Second derivative
test– A method to find the local extrema of a function by using the second derivative to analyze the
critical values for 𝑓(𝑥)
Calculus
v. 2015 - 2016
Content: Optimization
Duration: February (2 Weeks)
Essential What techniques can be used to find the absolute maximum or minimum value of a function?
Question: How can a real world problem be solved using absolute extrema?



Understand the extreme value theorem.
Find the absolute extrema for a given function on a closed interval and open interval.
Setup and solve real world optimization problems using the proper steps.
1. Find the primary equation.
2. Find a secondary equation (if needed)
3. Write the primary equation in terms of one variable.
4. Use the proper techniques of calculus to find the desired maximum or minimum value.
5. Answer the question.



Find the absolute extrema for the function 𝑓(𝑥) = 𝑥 2 (3 − ln 𝑥)on the interval (0, ∞).
Find the absolute extrema for the function 𝑓(𝑥) = √𝑥 2 + 𝑥 + 12 on the interval [−2,5]
Find two positive numbers such that the sum of the first and three times the second is 38 and the
product is maximized.
A Norman window has the shape of a rectangle topped with a semi-circle. (Thus the diameter of
the semi-circle is equal to the width of the rectangle.) If the perimeter of the window is 30 feet, find
the dimensions (radius and height) of the window so that the greatest possible amount of light is
admitted.
A man is at point A on the bank of a straight river, 3 km wide, and wants to reach point B, 8 km
downstream on the opposite bank, as quickly as possible. He could row to point C and then run to
point B, or he could row directly to point B, or he could row to some point D between C and B and
then run to point B. If he can row at 6 km/h and run at 8 km/h, where should he land to reach be
as soon as possible?
A rectangular storage container with an open top is to have a volume of 10 m 3. The length of the
base is twice the width. Material for the base cost $ 10 per square meter. Material for the sides costs
$ 6 per square meter. Find the cost of the materials for the cheapest such container.
Given the following cost function; C(x) = 0.001x3 – 5x + 250; where the cost is in dollars and x is
the number of units produced, find the minimum average cost per unit.
Skill:

Assessment: 


Resources:
Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Section 4.5
(pages 292 – 308)
Calculus: AP Edition. Pearson: Sections 4.4 (pages 281 – 291)
ACTIVITY: Pop Can Optimization (based on # 45 on page 288 of AP Edition. Pearson)
Standards: This content is beyond the scope of the PA Core Mathematical Standards.
Calculus
Vocabulary:
Comments:
v. 2015 - 2016
Absolute maximum– The highest point on the graph of a function. There is an absolute maximum a
𝑥 = 𝑐 if 𝑓(𝑐) ≥ 𝑓(𝑥) for all 𝑥; Absolute minimum– The lowest point on the graph of a function.
There is an absolute minimum a 𝑥 = 𝑐 if 𝑓(𝑐) ≤ 𝑓(𝑥) for all 𝑥; Extreme value theorem– If a function
is continuous on an interval, the function will have an absolute maximum and absolute minimum on that
interval; Primary equation– The equation that needs to maximized or minimized.
Calculus
v. 2015 - 2016
Content: Anti-differentiation
Duration: February/March (3 Weeks)
Essential If you know the derivative of a function, how can you work backwards to find the original function?
Question:




Understand the relationship between the derivative and the anti-derivative.
Know the difference between the antiderivative and indefinite integral.
Recognize that there are an infinite number of anti-derivatives for a function and graphically
interpret this idea.
Know and understand the rules for integration.
o ∫ 𝑘 𝑑𝑥 = 𝑘𝑥 + 𝑐 when 𝑘 is a constant
𝑥 𝑛+1
Skill:











Assessment: 
o ∫ 𝑥 𝑛 𝑑𝑥 =
+ 𝑐 except when 𝑛 = −1
𝑛+1
𝑥
𝑥
o ∫ 𝑒 𝑑𝑥 = 𝑒 + 𝑐
1
o ∫ 𝑑𝑥 = ln|𝑥| + 𝑐
𝑥
o ∫ 𝑘𝑓(𝑥) 𝑑𝑥 = 𝑘 ∫ 𝑓(𝑥)𝑑𝑥
o ∫ 𝑓(𝑥) 𝑑𝑥 ± ∫ 𝑔(𝑥) 𝑑𝑥 = ∫[𝑓(𝑥) ± 𝑔(𝑥)] 𝑑𝑥
Find the antiderivative of a function using the proper rules.
Understand the differential as a place holder for use with integration by substitution.
Understand the process of integration by substitution.
Use integration by substitution to evaluate an indefinite integral.
Know when it is appropriate to use integration by substitution.
Define Anti-differentiation and Integration.
Answer questions based on Integration properties and definitions..
o True/False: A constant factor can be moved across the integral sign.
o True/False: The integral of a constant function is zero.
o True/False: A function has only one anti-derivative.
Given the graph of one anti-derivative, find a possible graph for another anti-derivative of the same
function.
Given the graph of a function, find a possible graph for an anti-derivative.
Find a particular anti-derivative, given 𝑓(𝑥) and an initial condition.
Given 𝑓(𝑥) = 2𝑥 + ln 𝑥, find the differential 𝑑𝑦.
3
5𝑥 4
𝑥
3
Evaluate: ∫ ( +
) 𝑑𝑥

Evaluate: ∫(4𝑒 𝑥 + 𝑥 𝑒 − 𝑒𝑥) 𝑑𝑥

Evaluate: ∫
𝑥 3 +2𝑥−3
3𝑥
𝑥
𝑑𝑥

Evaluate: ∫

Evaluate: ∫


Evaluate: ∫ 𝑥 𝑒
𝑑𝑥
ln 𝑥
Evaluate: ∫
𝑑𝑥

Evaluate: ∫ 𝑥(3 − 𝑥)5 𝑑𝑥
𝑑𝑥
𝑥 2 +5
1+𝑒 −𝑥
𝑑𝑥
𝑒𝑥
2 3𝑥 2
𝑥
Calculus
Resources:
v. 2015 - 2016
Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Sections
6.1 & 6.2 (pages 363 – 389)
Calculus: AP Edition. Pearson: Sections 5.1 & 5.6 (pages 337 – 347 & 401 – 211)
Standards: This content is beyond the scope of the PA Core Mathematical Standards.
Vocabulary:
Comments:
Anti-derivative – The reconstruction of a function from its derivative; Differential – 𝑑𝑦 = 𝑓 ′ (𝑥) 𝑑𝑥;
Integral – The family of all anti-derivatives; Integral Symbol – ∫ ; Integrand – The function that is
being integrated. 𝑓(𝑥) in the integral ∫ 𝑓(𝑥) 𝑑𝑥; Variable of Integration – The independent variable
of the equation (𝑑𝑥).
Calculus
v. 2015 - 2016
Content: Definite Integrals
Duration: March (2 Weeks)
Essential How can the area under a curve be calculated?
Question:



Skill:
Define a definite integral (including an explanation of the mathematical formula).
Understand that a definite integral represents a signed area, and therefore can be positive, negative
or even zero.
Know and understand the properties of definite integrals.
𝑎
o ∫𝑎 𝑓(𝑥) 𝑑𝑥 = 0
𝑏
𝑎
o
∫𝑎 𝑓(𝑥) 𝑑𝑥 = − ∫𝑏 𝑓(𝑥) 𝑑𝑥
o
∫𝑎 𝑘𝑓(𝑥) 𝑑𝑥 = 𝑘 ∫𝑎 𝑓(𝑏) 𝑑𝑥 , where 𝑘 is a constant
o
∫𝑎 [𝑓(𝑥) ± 𝑔(𝑥)] 𝑑𝑥 = ∫𝑎 𝑓(𝑥) 𝑑𝑥 ± ∫𝑎 𝑔(𝑥) 𝑑𝑥
𝑏
𝑏
𝑏
𝑏
𝑏





Given a graph for 𝑓(𝑥) and the value of the areas on the graph, find the value of a given definite
integral using the definite integral definition and properties.
Given a function, an interval and the number or rectangles; use the Left, Right, Average and
Midpoint sum to approximate the area under the curve.
o

Assessment:


𝑏
o ∫𝑎 𝑓(𝑥) 𝑑𝑥 = ∫𝑎 𝑓(𝑥) 𝑑𝑥 + ∫𝑐 𝑓(𝑥) 𝑑𝑥, where 𝑎 < 𝑐 < 𝑏.
Approximate the area under a curve using Riemann Sums.
o Left (𝐿𝑛 )
o Right (𝑅𝑛 )
o Average (𝐴𝑛 )
o Midpoint (𝑀𝑛 )
Know and explain the Fundamental Theorem of Calculus.
Evaluate a definite integral using the Fundamental Theorem of Calculus.
o

𝑐
𝑏
y   x  5 From x = 0 to x = 5, using 4 intervals
From x = 1 to x = 2, using 5 intervals
y  x3
What is the best approximation to use for finding the area under the curve? How can all of the
approximation become more accurate?
Give a brief explanation of the following definition for a definite integral.
Let f be a continuous function on [a, b], and let
1. 𝑎 = 𝑥𝑜 < 𝑥1 < ⋯ , 𝑥𝑛−1 < 𝑥𝑛 = 𝑏
2. ∆𝑥𝑘 = 𝑥𝑘 − 𝑥𝑘−1 𝑓𝑜𝑟 𝑘 = 1,2, … 𝑛
3. ∆𝑥𝑘 → 0 𝑎𝑠 𝑛 → ∞
4. 𝑏𝑘−1 ≤ 𝑐𝑘 ≤ 𝑥𝑘 𝑓𝑜𝑟 𝑘 = 1,2, … , 𝑛
𝑏
Then ∫𝑎 𝑓(𝑥) 𝑑𝑥 = lim ∑𝑛𝑘=1 𝑓(𝑐𝑘 )∆𝑥𝑘
𝑥→∞
Explain the Fundamental Theorem of Calculus.
Use the Fundamental Theorem of Calculus to evaluate the definite integral of a function.
8 2
o ∫1 3 𝑑𝑥
3 √𝑥
2 2𝑥
2
o
∫1 ( 3 + 3𝑥) 𝑑𝑥
o
∫2 (𝑥 5−8) 𝑑𝑥
6
5𝑥 4
Calculus
Resources:
v. 2015 - 2016
Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Sections
6.4 & 6.5 (pages 401 – 431
Calculus: AP Edition. Pearson: Sections 5.2, 5.3 & 5.4 (pages 349 – 393)
Graphing Calculator
Standards: This content is beyond the scope of the PA Core Mathematical Standards.
Definite Integral– The cumulative sum of the signed areas bounded by a curve and the 𝑥-axis;
Vocabulary: Riemann Sum– A method for approximating the area under the curve using rectangles
Comments:
Calculus
v. 2015 - 2016
Content: Integral Applications
Duration: April (2 Weeks)
Essential How can integration be used to interpret the graph of a function?
Question:
Skill:




Define the average value of a function and explain using a graphical representation.
Find the average value of a function.
Find the area bounded by a curve and the x-axis.
Find the area bounded by two curves.

Given a function, find the average value of the function over a specified interval.
o 𝑦 = 𝑥 − 3𝑥 2 on the interval [−1,2]
o 𝑦 = √𝑥 + 1 on the interval [3,8]
o 𝑦 = 3𝑒 2𝑥 on the interval [2,5]
Given two functions, find the area bounded between them.
o 𝑦 = 𝑥 2 + 2 & 𝑦 = 𝑥 − 2 on the interval −1 ≤ 𝑥 ≤ 3
o 𝑦 = 𝑥 2 + 2𝑥 + 1 & 𝑦 = 2𝑥 − 5
o 𝑦 = 𝑥 −1 + 3 on the interval −4 ≤ 𝑥 ≤ −1
Assessment: 
Resources:
Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Sections
6.5 & 7.1 (pages 427 – 431 & 445 – 455)
Calculus: AP Edition. Pearson: Sections 5.5 & 6.2 (pages 393 – 400, 445 – 454)
Standards: This content is beyond the scope of the PA Core Mathematical Standards.
Average Value – The average height of the graph of 𝑓(𝑥) over the interval [𝑎, 𝑏]. As given by the
Vocabulary: formula: 𝑓 ̅ = 1 ∫𝑏 𝑓(𝑥) 𝑑𝑥
𝑏−𝑎 𝑎
Comments:
Calculus
v. 2015 - 2016
Content: Trigonometric Derivative and
Integrals
Duration: April (2 Weeks)
Essential How can the derivative and integral of a trigonometric function be calculated?
Question:

Know and understand the rules for differentiating trigonometric functions.
𝑑
o
sin 𝑥 = cos 𝑥
o
o
o
o
o

o
o
o
o




Assessment:



𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
cos 𝑥 = − sin 𝑥
tan 𝑥 = sec 2 𝑥
csc 𝑥 = − csc 𝑥 cot 𝑥
sec 𝑥 = sec 𝑥 tan 𝑥
cot 𝑥 = − csc 2 𝑥
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
cos 𝑓(𝑥) = −𝑓 ′ (𝑥) sin 𝑓(𝑥)
tan 𝑓(𝑥) = 𝑓 ′ (𝑥) sec 2 𝑓(𝑥)
csc 𝑓(𝑥) = −𝑓 ′ (𝑥) csc 𝑓(𝑥) cot 𝑓(𝑥)
sec 𝑓(𝑥) = 𝑓 ′ (𝑥) sec 𝑓(𝑥) tan 𝑓(𝑥)
cot 𝑓(𝑥) = −𝑓 ′ (𝑥) csc 2 𝑓(𝑥)
Know and understand the rules for integrating trigonometric function.
o ∫ sin 𝑥 𝑑𝑥 = − cos 𝑥 + 𝑐
o ∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝑐
o ∫ sec 2 𝑥 𝑑𝑥 = tan 𝑥 + 𝑐
o ∫ csc 𝑥 cot 𝑥 𝑑𝑥 = − csc 𝑥 + 𝑐
o ∫ sec 𝑥 tan 𝑥 𝑑𝑥 = sec 𝑥 + 𝑐
o ∫ csc 2 𝑥 𝑑𝑥 = − cot 𝑥 + 𝑐
Use integration by substitution to integrate trigonometric functions.
Use integration by substitution to develop 4 more trigonometric integration rules.
o ∫ tan 𝑥 𝑑𝑥 = − ln|cos 𝑥| + 𝑐
o ∫ sec 𝑥 𝑑𝑥 = ln|sec 𝑥 + tan 𝑥| + 𝑐
o ∫ cot 𝑥 𝑑𝑥 = ln|sin 𝑥| + 𝑐
o ∫ csc 𝑥 𝑑𝑥 = − ln|csc 𝑥 + cot 𝑥| + 𝑐
Evaluate the derivative of a trigonometric function.
cos 𝑥
1
o 𝑓(𝑥) =
+
3
sin 𝑥
o 𝑓(𝑥) = sec 4 2𝑥 + 5
o 𝑓(𝑥) = 𝑒 (sin 𝑥)(cos 𝑥)
o 𝑓(𝑥) = ln|csc 𝑥 2 |
o

𝑑𝑥
𝑑
Use the trigonometric rules for differentiation in conjunction with prior rules (especially how the
chain rule relates).
𝑑
o
sin 𝑓(𝑥) = 𝑓 ′ (𝑥) cos 𝑓(𝑥)
o
Skill:
𝑑𝑥
𝑑
𝑓(𝑥) =
Evaluate: ∫ (
𝑒 3𝑥
2 tan 𝑥
cos 𝑥
4
(sec 2
− 3 sin 𝑥) 𝑑𝑥
Evaluate: ∫
𝑥 + tan 𝑥) 𝑑𝑥
Evaluate: ∫(cos 2𝑥 − csc 𝑥) 𝑑𝑥
Evaluate: ∫ sin2 𝑥 cos 𝑥 𝑑𝑥
Calculus
Resources:
v. 2015 - 2016
sin 𝑒 −𝑥

Evaluate: ∫

Evaluate: ∫ tan 𝑥 sec 3 𝑥 𝑑𝑥

Evaluate: ∫
𝑒𝑥
𝑑𝑥
(cos 𝑥)[ln|sin 𝑥|]
sin 𝑥
𝑑𝑥
Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Sections
9.2 & 9.3 (pages 566 – 579)
Calculus: AP Edition. Pearson: Sections 3.5, 5.1 & 5.6 (pages 173 – 181, 341, 405)
Standards: This content is beyond the scope of the PA Core Mathematical Standards.
Vocabulary: No new vocabulary
Comments:
Calculus
v. 2015 - 2016
Content: Review of Trigonometric
Functions
Duration: May (1 Week)
do you find exact values for the six trigonometric functions of special angles?
Essential How
What is the difference between an identity and an equation?
Question: How do the graphs of the six trigonometric functions compare?

Skill: 





Evaluate the EXACT value of the sine, cosine, tangent, secant, cosecant or cotangent at a given
angle in radian measure. (Using the unit circle)
Know and use the trigonometric identities to manipulate and solve an equation.
Graph the six trigonometric functions.
Use transformations to graph trigonometric functions.
Completely fill in a unit circle or table of trigonometric values.
Use the unit circle or table of trigonometric values to evaluate the sine, cosine, tangent, secant,
cosecant or cotangent of an angle.
sin ( )
o
tan ( )
o
sec (−
o
o

3
5𝜋
6
11𝜋
6
)
Use the Reciprocal, Pythagorean, Double and Half angle identities to prove trigonometric equations.
o
Assessment:
2𝜋
o
csc2 𝑥−1
csc2 𝑥
sin 𝑥
1−cos 𝑥
2
= cos 2 𝑥
= csc 𝑥 + cot 𝑥
sin 𝑥 =
1−cos 2𝑥
2
Use the Reciprocal, Pythagorean, Double and Half angle identities to solve trigonometric equations.
o 2𝜃 cos 𝜃 + 𝜃 = 0
o 2 − 2 cos 2 𝑥 = sin 𝑥 + 1
𝑥
o sin2 𝜃 = 2 cos 2
2
o


1
2
Graph the sine, cosine, tangent, secant, cosecant and cotangent functions.
Use transformations to graph trigonometric functions.
o Graph 𝑦 = 2 sin 3𝑥 + 2
o
Resources:
cos 2 𝑥 − cos 2𝑥 =
𝜋
Graph 𝑦 = − cos (𝑥 + )
2
Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Section 9.1
(pages 557 – 566)
Calculus: AP Edition. Pearson: Section 1.4 (pages 41 – 53)
Calculus
v. 2015 - 2016
Standards: This content is beyond the scope of the PA Core Mathematical Standards.
Amplitude– The distance from the mean of a function to the maximum or minimum value; Double
Angle Identity– sin 2𝑥 = 2 sin 𝑥 cos 𝑥 or cos 2𝑥 = cos 2 𝑥 − sin2 𝑥 ; Half Angle Identity–
1−cos(2𝑥)
1+cos(2𝑥)
or cos 2 𝑥 =
; Identity– An equation that holds true for all real numbers;
Vocabulary: sin2 𝑥 =
2
2
Period– The distance it takes for a function to complete one full cycle; Pythagorean Identity–
sin2 𝑥 + cos 2 𝑥 = 1
Comments:
Calculus
v. 2015 - 2016
Content: Derivative Applications of
Trigonometric Functions
Duration: May (2 Weeks)
Essential How can the graph of a trigonometric function be explained/interpreted using techniques of calculus?
Question:








Write the equation of a tangent line at a given point for a trigonometric function.
Determine where the graph of a trigonometric function has a horizontal tangent line.
Determine where the graph of a trigonometric function is increasing, decreasing and locate the
local extrema (First Derivative Test).
Determine where the graph of a trigonometric function is concave up, concave down and locate
inflection points.
Use the Second Derivative Test to find the local extrema of a function.
Determine the absolute extrema of a trigonometric function on a closed interval.
Solve Related Rate problems involving trigonometric functions.
Solve Optimization problems involving trigonometric functions.
Evaluate definite integrals involving trigonometric functions.
Find the area bounded between two trigonometric functions.

Write the equation of a tangent line to 𝑦 = 4 + cot 𝑥 − 2 csc 𝑥, at 𝑥 = .



What is the equation of a line tangent to 𝑥 2 cos 2 𝑦 − sin 𝑦 = 0 at (0, 𝜋).
Where does 𝑔(𝑥) = sin2 𝑥, have horizontal tangent lines? (Give all answers between 0 and 2𝜋)
Where is the function 𝑓(𝑥) = 𝑥 + sin 𝑥, increasing and decreasing. Where does 𝑓(𝑥) have local
extrema? (Give all answers between 0 and 2𝜋)
Where is ℎ(𝑥) = 𝑒 −𝑥 sin 𝑥, concave down and concave up. Where does ℎ(𝑥) have inflection
points? (Give all answers between 0 and 2𝜋)
Use the second derivative test to locate the local extrema for the function 𝑓(𝑥) = sin 𝑥 − cos 𝑥
(Give all answers between 0 and 2𝜋)
A 13 foot ladder is leaning against a house, when its base starts to slide away. By the time the base
is 12 feet from the house, the base is moving at a rate of 5 feet per second. At what rate is the
angle between the ladder and the ground changing at that moment?
The trough in the figure is to be made to the dimensions shown. Only the angle 𝜃 can be varied.
What value of 𝜃 will maximize the volume of the trough?

Skill: 

Assessment:



Resources:
𝜋
2
Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Section
9.2 (pages 566 – 572)
Calculus: AP Edition. Pearson: Sections 3.11, 4.1, 4.2 & 4.4 (pages 232 – 240, 246 – 270 & 281 – 291)
Standards: This content is beyond the scope of the PA Core Mathematical Standards.
Calculus
Vocabulary: No new vocabulary
Comments:
v. 2015 - 2016
Calculus
v. 2015 - 2016
Content: Integration Applications of
Trigonometric Functions
Duration: May/June (2 Weeks)
Essential How can integration be used to interpret a function and its graph?
Question:
Skill:
Assessment:
Resources:




Evaluate a trigonometric definite integral.
Find the area bounded by the graph of a trigonometric function and the 𝑥-axis.
Find the area bounded by two trigonometric functions.
Find average value of a trigonometric function.

Evaluate ∫𝜋 cos 𝑥 𝑑𝑥

Evaluate ∫04

Evaluate



Find the area bounded by 𝑦 = 𝑥 sin 𝑥 2 and the 𝑥-axis between 𝑥 = 0 and 𝑥 = √𝜋
Find the area bounded by 𝑦 = sin 𝑥 cos 𝑥 and 𝑦 = 0 between 𝑥 = 0 and 𝑥 = √𝜋
Find the area bounded by 𝑦 = sin 𝑥 and 𝑦 = sin 2𝑥 on the interval [0, 𝜋]

Find the average value of the function 𝑓(𝑥) = sec 3 𝑥 tan 𝑥 on the interval [0, ]
2𝜋
3
𝜋
sin 𝑥
cos2 𝑥
𝜋
𝑥
∫02 tan 2
𝑑𝑥
𝑑𝑥
3
Calculus: For Business, Economics, Life Sciences, and Social Sciences. 9th ed. Prentice Hall; Section 9.3
(pages 572 – 577)
Calculus: AP Edition. Pearson: Sections 5.3 – 5.5 & 6.2 (pages 364 – 400 & 445 – 454)
Standards: This content is beyond the scope of the PA Core Mathematical Standards.
Vocabulary: No new vocabulary
Comments:
𝜋