UCHS AP Calculus BC

AP Exam Date:  Tuesday, May 9, 2017

AP Exam Time:  8:00 am

AP Exam Location:  First Baptist Church

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Exam Review #5   (#’s 1 – 10 only) Due Date:  Wednesday 2/8

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what-is-chapter7-bankruptcyTechniques of Integration

Content Vocabulary

Integrable Function – A function for which the definite integral exists. Piecewise

continuous functions are integrable, and so are many functions that are not piecewise continuous.

Antiderivative of a Function – A function that has a given function as its derivative.

Integral of a Function – The result of either a definite integral or an indefinite integral.

Indefinite integral – The family of functions that have a given function as a common derivative. The indefinite integral of f(x) is written ∫ f(x) dx.

Fundamental Theorem of Calculus (FTC) – The theorem that establishes the connection between derivatives, antiderivatives, and definite integrals.

Integrand – The function being integrated in either a definite or indefinite integral.

Definite integral – An integral which is evaluated over an interval. A definite integral is written . Definite integrals are used to find the area between the graph of a function and the x-axis. There are many other applications.

Limits of Integration – For the definite integral , the bounds (or limits) of integration are a and b.

U Substitution – An integration method that essentially involves using the chain rule in reverse.

Integration by Parts – A formula used to integrate the product of two functions.

Partial Fractions – The process of writing any proper rational expression as a sum of proper rational expressions.

Trigonometric Substitution – A method for computing integrals often used when the integrand contains expressions of the form a2 – x2, a2 + x2, or x2 – a2.

Suggested Textbook  Problems


Textbook Solutions

7.1 Solutions

7.2 Solutions

7.3 Solutions

7.5 Solutions

7.6 Solutions

7.8 Solutions

Chapter 7 Review Exercises Solutions 



Differential Equations


Content Vocabulary

Differential Equation – An equation showing a relationship between a function and its derivative(s).

Euler’s Method – a numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.

Ordinary Differential Equation – a differential equation containing one or more functions of one independent variable and its derivatives

Logistic Differential Equation –  (sometimes called the Verhulst model or logistic growth curve) is a model of population growth



Textbook Solutions

9.1 Solutions

9.3 Solutions

9.4 Solutions



Infinite Series

Content Vocabulary

Sequence – A list of numbers set apart by commas, such as 1, 3, 5, 7, . . .

Bounded Sequence – A sequence with terms that have an upper bound and a lower bound.

Series – The sum of the terms of a sequence.

Alternating Series – A series which alternates between positive and negative terms.

Converge – To approach a finite limit.

Diverge – To fail to approach a finite limit.

Convergent Series – An infinite series for which the sequence of partial sums converges.

Divergent Series – A series that does not converge.

Convergent Sequence – A sequence with a limit that is a real number.

Divergent Sequence – A sequence that does not converge.

Limit Test for Divergence – A convergence test that uses the fact that the terms of a convergent series must have a limit of zero.

Integral Test – A convergence test used for positive series with decreasing terms.

Comparison Test – A convergence test which compares the series under consideration to a known series. Essentially, the test determines whether a series is “better” than a “good” series or “worse” than a “bad” series. The “good” or “bad” series is often a p-series.

Limit Comparison Test – A convergence test often used when the terms of a series are rational functions. Essentially, the test determines whether a series is “about as good” as a “good” series or “about as bad” as a “bad” series. The “good” or “bad” series is often a p-series.

Ratio Test – A convergence test used when terms of a series contain factorials and/or nth powers.

Root Test – A convergence test used when series terms contain nth powers.

Power Series – A series which represents a function as a polynomial that goes on forever and has no highest power of x.

Derivative of a Power Series – The derivative of a function defined by a power  series can be found by differentiating the series term-by-term.

Remainder of a Series – The difference between the nth partial sum and the sum of a series.

Infinite Series – A series that has no last term.

Infinite Geometric Series – An infinite series that is geometric. An infinite geometric series converges if its common ratio r satisfies –1 < r < 1. Otherwise it diverges.

Interval of Convergence – For a power series in one variable, the set of values of the variable for which the series converges. The interval of convergence may be as small as a single point or as large as the set of all real numbers.

Nth Degree Taylor Polynomial – An approximation of a function using terms from the function’s Taylor series. An nth degree Taylor polynomial uses all the Taylor series terms up to and including the term using the nth derivative.

Nth Partial Sum – The sum of the first n terms of an infinite series.

P-Series – A series of the form or , where p > 0. Often employed when using the comparison test and the limit comparison test.

Taylor Series – The power series in x – a for a function f .

Maclaurin Series – The power series in x for a function f(x).

Taylor Series Remainder – A quantity that measures how accurately a Taylor  polynomial estimates the sum of a Taylor series.


Suggested Textbook  Problems


Textbook Solutions

10.1 Solutions

10.2 Solutions

10.3 Solutions

10.4 Solutions

10.5 Solutions

10.6 Solutions

10.7 Solutions

Ch 10 Chapter Review Solutions

Ch 10 AP Prep Solutions




chapter_11Parametric Equations


Polar Coordinates


Content Vocabulary

Parametric Equation – A system of equations with more than one dependent variable. Often parametric equations are used to represent the position of a moving point.

Parameter – The independent variable or variables in a set of parametric equations.

Parametrize – To write in terms of parametric equations.

Polar Coordinates – A way to describe the location of a point on a plane. A point is given coordinates (r, θ). r is the distance from the point to the origin. θ is the angle measured counterclockwise from the polar axis to the segment connecting the point to the origin.

Spiral – A curve on a plane that turns endlessly outward or inward (or both). Spirals usually have polar equations.

Lemniscate – A curve usually expressed in polar coordinates that resembles a figure eight.

Cardioid – A curve that is somewhat heart shaped. A cardioid can be drawn by tracing the path of a point on a circle as the circle rolls around a fixed circle of the same radius. The equation is usually written in polar coordinates.

Limacon – A famliy of related curves usually expressed in polar coordinates. The cardioid is a special kind of limaçon.

Polar Rose Curve – A smooth curve with leaves arranged symmetrically about a common center.

Arc LengthThe length of a curve or line.


Suggested Textbook  Problems


Textbook Solutions

11.1 Solutions

11.2 Solutions

11.3 Solutions

11.4 Solutions

11.5 Solutions

11.6 Solutions

11.7 Solutions

Ch 11 Review Solutions