AP Calculus AB & BC Crash Course

Flashcards for AP Calculus AB & BC exam preparation covering key vocabulary and concepts from the provided lecture notes.

AP Calculus AB & BC Crash Course

A targeted test prep designed to assist you in your preparation for either version of the AP Calculus exam.

Targeted Review Chapters

Highlights topics that are exclusive to the BC version of the test.

Basic Functions

Consists of Polynomials, absolute value, and square root functions.

y=sin(x)

x-intercepts: x = nπ, n is an integer; y-intercept: y=0; Odd

y = cos(x)

x-intercepts: x = (2n+1) * (π/2), n is an integer; y-intercept: y=1; Even

y = tan(x)

x-intercepts: x = nπ, n is an integer; vertical asymptotes: x = (2n+1)*(π/2), n is an integer; y-intercept: y=0; Odd

y = sin-1(x)

Domain: [-1, 1] Range: [-π/2, π/2]; Strictly increasing

y = cos-1(x)

Domain: [-1, 1]; Range: [0, π]; Strictly decreasing

y = tan-1(x)

Domain: (-∞, ∞); Range: (-π/2, π/2); Horizontal Asymptotes: y = ±π/2; strictly increasing

y = ex

This is the inverse of y = ln(x); Strictly increasing; x-intercepts: none; y-intercept: y=1; horizontal asymptote: y = 0

y = ln(x)

This is the inverse of y = ex; Strictly increasing; x-intercept: x=1; y-intercepts: none; vertical asymptote: x = 0

y=1/x

Undefined at x = 0; Not differentiable at x = 0; x-intercepts: none; y-intercepts: none; horizontal asymptote: y = 0; vertical asymptote: x = 0; lim (x->0) f(x) dne (does not exist)

y=1/x^2

Undefined at x = 0; Not differentiable at x = 0; x-intercepts: none; y-intercepts: none; horizontal asymptote: y = 0; vertical asymptote: x = 0; lim (x->0) f(x) = ∞

Limit of a Function

Where the limit of f(x) as x approaches a number or ± ∞, represents the value that y approaches.

Left Hand Limit

States that as x approaches a, from the left of a, f(x) approaches L.

Right Hand Limit

States that as x approaches a, from the right of a, f(x) approaches L.

Limit of a function at a point

States that as x approaches a, simultaneously from the left and right of a, f(x) approaches L.

Asymptotes

Are vertical or horizontal lines (the AP Calculus exams do not include oblique asymptotes) which a graph approaches.

Vertical Asymptotes

Are vertical lines that a graph only approaches but never intersects.

Horizontal Asymptotes

Are horizontal lines that a graph approaches and may intersect.

Unbounded Behavior

If a function, y = f(x), approaches positive infinity either as x → a or as x → ±∞, the function is said to increase without bound.

Continuity of Functions

A function is either continuous (no breaks whatsoever) or discontinuous at certain points.

Removable Discontinuity

Occurs when an otherwise continuous graph has a point (or more) missing.

Nonremovable Discontinuity

Occurs at step breaks in the graph or at vertical asymptotes.

Intermediate Value Theorem

If f is continuous on [a, b] and c is a number satisfying f(a) ≤ c ≤ f(b), then there is at least one number x in [a, b], such that f(x) = c.

Parametric and Vector Equations

Are used to describe the motion of a body. They have different notations but describe the same concept. An additional variable is involved, called the parameter, usually denoted by t (for time).

Polar Equation

Is written using polar coordinates (r, θ), where r represents the point’s distance from the origin, and θ represents the measurement of the angle between the positive x-axis and the line segment between the point and the origin.

Polar coordinates (r, θ)

Represents the point’s distance from the origin, and θ represents the measurement of the angle between the positive x-axis and the line segment between the point and the origin

Meaning of Derivative

The derivative of a function is its slope. A linear function has a constant derivative since its slope is the same at every point.

Local Linearity or Linearization

When asked to find the linearization of a function at a given x- value or when asked to find an approximation to the value of a function at a given x-value using the tangent line, this means finding the equation of the tangent line

Common terms to describe derivative

Also called instantaneous rate of change, change in y with respect to x, or slope.

L’Hôpital’s Rule

Allows you to take limits that have indeterminate forms, such as 0/0 or ∞/∞.

Product Rule

(f(x)g(x))′ = f′(x)g(x) + f(x)g′(x)

y = ex

= ex!!!. It is the only function which is equal to its derivative.

Implicit Differentiation

this means finding y′ when the equation given is not explicitly defined in terms of y (that is, it is not of the form y = f(x)).

Derivatives and Intervals of Increase and Decrease

If f′(x) > 0 on (a, b) then f(x) is increasing on (a, b).

Critical Points of f(x)

Points in its domain at which f′(x) = 0 or f′(x) does not exist.

Derivatives and Concavity

If f″(x) > 0 on (a, b), then f(x) is concave up on (a, b).

Inflection Point

A point at which the concavity of f(x) changes

x-intercept

A point at which a function intersects the x-axis, and hence, y = 0 here.

y-intercept

A point at which a function intersects the y-axis, and hence, x = 0 here.

Relative Maximum Point

A point on a function is a relative (or local) maximum point if and only if it is the highest point in its neighborhood.

Relative Minimum Point

A point on a function is a relative (or local) minimum point if and only if it is the lowest point in its neighborhood.

Absolute Maximum Point

A point on a function is an absolute maximum point if and only if it is the highest point.

The Mean Value Theorem (MVT)

States that if a function is continuous on [a, b] and differentiable on (a, b) then there exists at least one x value, x = c, where a < c < b, such that

Rolle’s Theorem

States that if a function is continuous on [a, b] and differentiable on (a, b) and f(a) = f(b) then, there exists an x value, x = c, where a < c < b, such that f′(c) = 0.

Newton’s Method

The concept of derivative is used to find the roots of a function.

Euler’s Method

This method is used for approximating values of a function given a point on the function, the function’s derivative and the step size for x (the smaller the step size, the better the approximations).

Indefinite Integrals

Have no limits, ∫f(x)dx. This represents the antiderivative of f(x).

Definite Integrals

Have limits x= a and x= b, If f(x) is continuous on [a, b] and F′(x) = f(x), then

Improper Integrals

Have one or both limits equal to either positive or negative infinity or are discontinuous on the given interval.

The First Fundamental Theorem of Calculus

States that if f(x) is continuous on [a, b] and F′(x) = f(x), then .

The Second Fundamental Theorem of Calculus

States that if f(x) is continuous on [a, b], then .

The Mean Value Theorem for Integrals

if f(x) is continuous on [a, b] then there is a c in [a, b], such that .

Average Value of a Function

If f(x) is continuous on [a, b], the average value of f(x) on [a, b] is given by

Riemann Sums (LRAM, RRAM, MRAM)

Are used to approximate the area between a function and the x-axis by slicing the area into thin vertical rectangles.

LRAM—Left Rectangle Approximation Method

To approximate the area between a function, f(x) and the x-axis on [a, b], slice the area into vertical rectangular strips each of width Δx (the value of Δx will be given in the problem). Starting on the left, create rectangles and add up all their areas.

RRAM—Right Rectangle Approximation Method

Using the function f(x), above, create rectangles starting on the right such that the upper right corner of each rectangle is on the curve.

MRAM—Midpoint Rectangle Approximation Method

Using the function f(x), above, create rectangles such that the height of each rectangle is in the middle and the midpoint of the upper width of each rectangle is on the curve.

Trapezoid Rule

Given the function f(x), as above, connect the top endpoints of the vertical line segments, thus creating trapezoids. Add up the areas of all the trapezoids. This is an approximation of the area between the curve and the x-axis.

Optimization

Optimizing a quantity means to find its maximum or minimum point.

Related Rates

You take the derivative implicitly, with respect to time.

U-Substitution

Is used to rewrite the integrand so that it is easily integrable. This method is used when the integrand is of the form f(g(x))g′(x) where g′(x) can be off by a constant factor.

Integration by Parts

Is used when the integrand is a product of unrelated functions, of the form f(x)g’(x).

Integration by Partial Fractions

Is used by separating a fraction into partial fractions.

Sequences

A list of numbers separated by commas a1, a2, a3,…, ak,…, that may or may not have a pattern.

Series

a series is the sum of the terms of a sequence.

Geometric Series

This series is of the form a + ar + ar2 + ··· + arn + …. This series converges (that is, its sum exists) if and only if |r| < 1 (that is, -1 < r < 1).

p-series

, p > 0, converges when p > 1 and diverges when 0 < p ≤ 1.

Alternating Series

Are series with terms whose signs alternate. They are of the form .

Harmonic Series

Diverges

Alternating Harmonic series

Converges

Alternating p-series

converges for p > 0.

Telescoping Series

A series in which all but a finite number of terms cancel out.

Divergence Test

If ≠ 0, the series diverges.

Absolute Convergence

If a series converges absolutely, then it converges.

Ratio test for absolute convergence

If series need not have positive terms and need not be alternating to use this test.

Limit Comparison Test

Series satisfy the formula Lim k->inf ak / bk = c , where 0 < c < inf , then the series converges/diverges

Power Series

Power Series exactly one of the following holds:

Taylor Series

Power Series used to to find the equations of tangent lines to functions to see how well polynomials can approximate functions.