Section 2.1 Linear Functions
... 1. Determine the leading term. Is the degree even or odd? Is the leading coefficient positive or negative? Use the answers to both questions to determine the end behavior. 2. Find the y-intercept. 3. Factor the polynomial. 4. Find the x-intercept(s). 5. Plot the x-intercepts and y-intercept on the 2 ...
... 1. Determine the leading term. Is the degree even or odd? Is the leading coefficient positive or negative? Use the answers to both questions to determine the end behavior. 2. Find the y-intercept. 3. Factor the polynomial. 4. Find the x-intercept(s). 5. Plot the x-intercepts and y-intercept on the 2 ...
randolph township school district
... The Hillside Township School District is committed to excellence. We believe that all children are entitled to an education that will equip them to become productive citizens of the twenty-first century. We believe that a strong foundation in mathematics provides our students with the necessary skil ...
... The Hillside Township School District is committed to excellence. We believe that all children are entitled to an education that will equip them to become productive citizens of the twenty-first century. We believe that a strong foundation in mathematics provides our students with the necessary skil ...
Slide
... In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not co ...
... In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not co ...
Math 2415 – Calculus III Calculus
... Math 2415 – Calculus III Calculus: Early Transcendentals, 8th ed. Alternate Edition with EWA, James Stewart Brooks Cole; 8th edition; ISBN-13: 978-1285741550 ...
... Math 2415 – Calculus III Calculus: Early Transcendentals, 8th ed. Alternate Edition with EWA, James Stewart Brooks Cole; 8th edition; ISBN-13: 978-1285741550 ...
Solution - Harvard Math Department
... The following two pictures show bifurcation diagrams. The vertical axes is the deformation parameter c. On the left hand side, we see the bifurcation diagram of the function f (x) = x6 − x4 + cx2 , on the right hand side the bifurcation diagram of the function f (x) = x5 − x4 + cx2 . As done in cla ...
... The following two pictures show bifurcation diagrams. The vertical axes is the deformation parameter c. On the left hand side, we see the bifurcation diagram of the function f (x) = x6 − x4 + cx2 , on the right hand side the bifurcation diagram of the function f (x) = x5 − x4 + cx2 . As done in cla ...
Lesson 3-8: Derivatives of Inverse Functions, Part 1
... Let f x x5 3x 2 , and let f 1 denote the inverse of f . Given that 1, 2 is on the graph of f, find ...
... Let f x x5 3x 2 , and let f 1 denote the inverse of f . Given that 1, 2 is on the graph of f, find ...
Pre calculus Topics
... Linear approximation. Interpretations and properties of definite integrals. Instantaneous rate of change Area between curves Average rate of change. ...
... Linear approximation. Interpretations and properties of definite integrals. Instantaneous rate of change Area between curves Average rate of change. ...
Unit 3. Integration 3A. Differentials, indefinite integration
... 3F-8 a) Find all plane curves such that the tangent line at P intersects the x-axis 1 unit to the left of the projection of P on the the x-axis. b) Find all plane curves in the first quadrant such that for every point P on the curve, P bisects the part of the tangent line at P that lies in the first ...
... 3F-8 a) Find all plane curves such that the tangent line at P intersects the x-axis 1 unit to the left of the projection of P on the the x-axis. b) Find all plane curves in the first quadrant such that for every point P on the curve, P bisects the part of the tangent line at P that lies in the first ...
continuity
... In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not co ...
... In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not co ...
Calculus Challenge #7 SOLUTION
... Must the interval of integration be [0, 1] for polynomials to be non-fundamental? Can we find a more general result? No, [0, 1] is convenient, but any other interval will do. ...
... Must the interval of integration be [0, 1] for polynomials to be non-fundamental? Can we find a more general result? No, [0, 1] is convenient, but any other interval will do. ...
Department of Physics and Mathematics
... 15. Understand the notion of “area under the curve”. 16. Find anti-derivatives using the power rule and u-substitution. 17. Compute the derivative and anti-derivative of trigonometric functions. 18. State and apply the Fundamental Theorem of Calculus, the Mean Value Theorem for Derivatives, the Mean ...
... 15. Understand the notion of “area under the curve”. 16. Find anti-derivatives using the power rule and u-substitution. 17. Compute the derivative and anti-derivative of trigonometric functions. 18. State and apply the Fundamental Theorem of Calculus, the Mean Value Theorem for Derivatives, the Mean ...
Week 3. Functions: Piecewise, Even and Odd.
... numbers x ∈ R such that f(x) makes sense.) • What is the range of f? (The range of f is the set of all numbers ...
... numbers x ∈ R such that f(x) makes sense.) • What is the range of f? (The range of f is the set of all numbers ...
trigint - REDUCE Computer Algebra System
... 1+u2 1+u2 1+u2 There are of course, other trigonometric substitutions, used by REDUCE, such as sin and cos, but since these are never singular, they cannot lead to problems with discontinuities. Given an integrable function f (sin x, cos x) whose indefinite integral is required, select one of the su ...
... 1+u2 1+u2 1+u2 There are of course, other trigonometric substitutions, used by REDUCE, such as sin and cos, but since these are never singular, they cannot lead to problems with discontinuities. Given an integrable function f (sin x, cos x) whose indefinite integral is required, select one of the su ...
Section 4.3 Line Integrals - The Calculus of Functions of Several
... done by a force when an object moves along the curve should depend only on the curve and not on any particular parametrization of the curve. We need to verify the previous statement in general before we can state our definition of the line integral. Note that in these two examples, ψ(t) = ϕ 2t . In ...
... done by a force when an object moves along the curve should depend only on the curve and not on any particular parametrization of the curve. We need to verify the previous statement in general before we can state our definition of the line integral. Note that in these two examples, ψ(t) = ϕ 2t . In ...
aCalc02_3 CPS
... So to prove continuity at x=c we have to: 1. Prove that the limit exists… so we know that the function approaches the same value from both values larger and values lower than the value we want to prove continuity for. ...
... So to prove continuity at x=c we have to: 1. Prove that the limit exists… so we know that the function approaches the same value from both values larger and values lower than the value we want to prove continuity for. ...
Computing Derivatives and Integrals
... This is because an indefinite integral really represents an antiderivative - and antiderivatives are not unique! Indeed, not only is sin(x) an antiderivative of cos(x), but so is sin(x) + 25. The +c accounts for all of the possible different antiderivatives. Remark 1.3. Note that there are a few “fa ...
... This is because an indefinite integral really represents an antiderivative - and antiderivatives are not unique! Indeed, not only is sin(x) an antiderivative of cos(x), but so is sin(x) + 25. The +c accounts for all of the possible different antiderivatives. Remark 1.3. Note that there are a few “fa ...
Math 500 – Intermediate Analysis Homework 8 – Solutions
... 0, if x = 0 f (x) = 1, if x > 0 on the set [0, ∞). (b) Does fn → f uniformly on [0, 1]? Explain. Solution: Notice that although each function fn is continuous on [0, 1], the pointwise limit f is not continuous on [0, 1]. Thus, the convergence can not be uniform by Theorem 24.3. (c) does fn → f unifo ...
... 0, if x = 0 f (x) = 1, if x > 0 on the set [0, ∞). (b) Does fn → f uniformly on [0, 1]? Explain. Solution: Notice that although each function fn is continuous on [0, 1], the pointwise limit f is not continuous on [0, 1]. Thus, the convergence can not be uniform by Theorem 24.3. (c) does fn → f unifo ...
Honors Precalculus Topics
... multiple-choice questions and free response questions. Partial credit may be awarded on some items. ...
... multiple-choice questions and free response questions. Partial credit may be awarded on some items. ...
AP Calculus AB Course Syllabus 2016-2017
... A. Perform advanced arithmetic and algebraic calculations. B. Differentiate algebraic and transcendental functions. C. Integrate algebraic and transcendental functions. D. Graph algebraic and transcendental functions. E. Apply calculus to real-life situations. F. Appreciate and understand that calcu ...
... A. Perform advanced arithmetic and algebraic calculations. B. Differentiate algebraic and transcendental functions. C. Integrate algebraic and transcendental functions. D. Graph algebraic and transcendental functions. E. Apply calculus to real-life situations. F. Appreciate and understand that calcu ...
∞ ∞ lnx sinx x =1 Local minimum
... 1. Read the problem carefully. Ask yourself: What is the unknown? What are the given quantities? What are the given conditions? In most cases it is useful to draw a diagram to better understand the problem. 2. Introduce the variables: Assign a variable to the quantity that is to be maximized or mini ...
... 1. Read the problem carefully. Ask yourself: What is the unknown? What are the given quantities? What are the given conditions? In most cases it is useful to draw a diagram to better understand the problem. 2. Introduce the variables: Assign a variable to the quantity that is to be maximized or mini ...
Week 2 - NUI Galway
... A function f is a rule that assigns to each element x in a set X exactly one element, called f(x), in a set Y. We write f: X → Y and say that “f is a function from X to Y”. Here, X is called the domain of the function f, and Y is called the codomain of f. The range of f is {f(x) : x ∈ X}, the set of ...
... A function f is a rule that assigns to each element x in a set X exactly one element, called f(x), in a set Y. We write f: X → Y and say that “f is a function from X to Y”. Here, X is called the domain of the function f, and Y is called the codomain of f. The range of f is {f(x) : x ∈ X}, the set of ...
Calculus Curriculum Questionnaire for Greece
... students in your country. If it is impossible to answer a particular question, just make a note and move to the next question. ...
... students in your country. If it is impossible to answer a particular question, just make a note and move to the next question. ...
CHAPTER SIX: APPLICATIONS OF THE INTEGRAL
... At times, we may have to curves that there is no “top” or “bottom” y, rather there are left and right curves. By definition, these curves would not be functions of x, since they would fail to honor the vertical line test (the test that determines if a relation is actually a function). But these func ...
... At times, we may have to curves that there is no “top” or “bottom” y, rather there are left and right curves. By definition, these curves would not be functions of x, since they would fail to honor the vertical line test (the test that determines if a relation is actually a function). But these func ...
answers, in pdf - People @ EECS at UC Berkeley
... given by a definite integral 0 2 − x(2 − x)dx = 0 2 − 2x + x2 dx = [2x − x2 + x3 /3]20 = 8/3 • 6.4 #10 Find the area of the region. Solution Note that y = e2x is an increasing function, and its value at x = 0 is 1. On the other hand, y = 1 − x is a decreasing function, and its value at x = 0 is also ...
... given by a definite integral 0 2 − x(2 − x)dx = 0 2 − 2x + x2 dx = [2x − x2 + x3 /3]20 = 8/3 • 6.4 #10 Find the area of the region. Solution Note that y = e2x is an increasing function, and its value at x = 0 is 1. On the other hand, y = 1 − x is a decreasing function, and its value at x = 0 is also ...
Lebesgue integration
In mathematics, the integral of a non-negative function can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.Mathematicians had long understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the area under the curve could be defined as the integral, and computed using approximation techniques on the region by polygons. However, as the need to consider more irregular functions arose—e.g., as a result of the limiting processes of mathematical analysis and the mathematical theory of probability—it became clear that more careful approximation techniques were needed to define a suitable integral. Also, we might wish to integrate on spaces more general than the real line. The Lebesgue integral provides the right abstractions needed to do this important job.The Lebesgue integral plays an important role in the branch of mathematics called real analysis, and in many other mathematical sciences fields. It is named after Henri Lebesgue (1875–1941), who introduced the integral (Lebesgue 1904). It is also a pivotal part of the axiomatic theory of probability.The term Lebesgue integration can mean either the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue—or the specific case of integration of a function defined on a sub-domain of the real line with respect to Lebesgue measure.