Calculus highlights for AP/final review

Evaluating Limits *Common methods for evaluating finite limits analytically (that don’t work with direct substitution): 1) Factor: 2) Rationalize:

(for indeterminate forms of continuous/differentiable functions) 3) Clearing Fractions: 4) L’Hopital’s Rule: (for indeterminate forms of continuous/differentiable functions)

limits at infinity and infinite limits *Evaluate limits of rational functions as x approaches infinity by using the rules of horizontal asymptotes:

Continuity, differentiability, and limits Limits: exists if Continuity: f(x) is continuous if: 1) exists 2) f(c) is defined 3)

Differentiability: f(x) is differentiable if exists. *If f(x) is differentiable, then it is continuous. *f(x) is not differentiable at any sharp turns, i.e., is not differentiable at x=0.

Which of the following are true for the following graph. I. exists for all values of c in the given domain II. f(x) is continuous on the given domain III. f(x) is differentiable on the given domain IV. f(c) is defined for all values of c in the given domain

Derivatives *Power Rule: *Product Rule: *Quotient Rule: *Chain Rule:

*Trigonometric Functions:

*Exponential and Logarithmic Functions:

*Implicit Differentiation: When differentiating a term of a function that contains y, multiply by y’ or

Find

tangent lines *The derivative or represents the slope of the graph of f(x) at any given value of x. *The equation of the tangent line to the graph of f(x) at a given point (x, y) is found by finding at that given value, then plugging it into the point-slope equation for a line.

Find the linear approximation of f(0.2) at x=0 for

average rate of change and instantaneous rate of change *Average Rate of Change/Approximate Rate of Change: (slope formula) *Instantaneous Rate of Change/Exact Rate of Change:

Intermediate value theorem, rolle’s theorem, and mean value theorem *Intermediate Value Theorem: If f(x) is continuous on [a, b] and f(a)<k<f(b), then there exists at least one value of c in [a, b] such that f(c)=k. *Rolle’s Theorem: If f(x) is continuous and differentiable, and f(a)=f(b), then there exists at least one value of c in (a, b) such that *Mean Value Theorem: If f(x) is continuous and differentiable then there exists a value c in (a, b) such that

related rates *Process for solving related rates problems: 1) Write an equation to represent the problem. 2) Find the derivative (implicitly, with respect to t) for the equation. 3) Plug in all known variables, including given rates. 4) Solve for the unknown.

Graphical analysis *Increasing/Decreasing Behavior: f(x) is increasing if f(x) is decreasing if *Concavity/Points of Inflection: f(x) is concave upward if f(x) is concave downward if f(x) has a point of inflection if changes sign

*Absolute/Global Extrema on [a, b]: 1) Find critical numbers (where or is undefined) 2) Plug in critical numbers and endpoints into f(x) 3) Find smallest y-value (absolute/global minimum) Find largest y-value (absolute/global maximum)

*Relative/Local Extrema: First Derivative Test: 1) Find critical numbers (where or is undefined) 2) When changes from positive to negative at a critical number, there is a relative maximum at (x, f(x)). When changes from negative to positive at a critical number, there is a relative minimum at (x, f(x)).

Second Derivative Test: 1) Find critical numbers 2) f(x) has a relative maximum if at a critical number. f(x) has a relative minimum if at a critical number .

For , find (a) the intervals on which f(x) is increasing or decreasing (b) the intervals on which f(x) is concave upward or concave downward (c) the points of inflection of f(x) (d) any relative extrema of f(x) (e) the absolute extrema of f(x) on [-2, 2]

Optimization *Process for solving optimization problems: 1) Draw and label a sketch, if applicable. 2) Write a primary equation to be optimized. 3) Use any secondary equations to rewrite the primary equation in terms of one variable. 4) Apply First Derivative Test, Second Derivative Test, or absolute extrema test to finding the maximum or minimum value.

Riemann sums and trapezoidal sums *Riemann Sum: Approximates the area under a curve with a finite number of rectangles that intercept the graph at their right endpoint, left endpoint, or midpoint. *Trapezoidal Sum: Approximates the area under a curve with a finite number of trapezoids.

Given the following table of values, find R(3), L(3), M(3), and T(3). x -1 2 7 10 f(x) 6 -4 Given , find R(4), L(4), M(4), and T(4) on the interval

Integrals *Definition of a Definite Integral: When the limit as the number of rectangles approaches infinity of a Riemann Sum is found, this represents the area under the curve bound by the x-axis, or the definite integral of the function on a given interval. A definite integral is also used to find the “total amount accumulated” of something given its rate of change.

*Power Rule: *U-Substitution: *Exponential and Logarithmic Functions:

*Trigonometric Functions:

*Special Properties of Integrals:

Fundamental theorems of calculus *First Fundamental Theorem of Calculus: -or- -or- (with U-Substitution)

*Second Fundamental Theorem of Calculus:

Given and F(3)=-10, find F(6).

average value of a function

The following graph shows the number of cellphone sales an AT&T representative makes at each hour of a given workday. Find the average number of sales the representative makes during [1, 8].

*Total Distance Traveled: particle motion *Position, Velocity, and Acceleration: *Speed: *Total Distance Traveled:

*A particle moves left when v(t)<0 and right when v(t)>0. *A particle changes direction when v(t) changes sign. *A particle stops when v(t)=0. *A particle speeds up when v(t) and a(t) are the same sign and slows down when v(t) and a(t) are opposite signs. *A particle is farthest to the left when x(t) is minimized and farthest to the right when x(t) is maximized.

solving differential equations through separation of variables *Differential equations are equations containing *To solve a differential equation means to integrate to find the original equation in terms of only x and y. We can do this by first separating the variables, then integrating both sides. Solve

Area between two curves *To find the area between two curves, integrate the difference between the larger function and the smaller function (top function - bottom function).

Volumes of revolution *Disk Method: To find the volume of the solid formed by revolving a region about a horizontal line that is adjacent to the region, use . To find the volume of the solid formed by revolving a region about a vertical line that is adjacent to the region, use .

*Washer Method: To find the volume of the solid formed by revolving a region about a horizontal line that is not adjacent to the region, use . To find the volume of the solid formed by revolving a region about a vertical line that is not adjacent to the region, use

Cross-sectional volumes *To find the volume of the solid formed by lying cross-sections in the form of a geometric shape perpendicular to the x-axis in a bounded region, use , where A is the area formula for the given geometric shape and f(x)-g(x) is the height of the representative rectangle in the region. Be sure you consider what quantity f(x)-g(x) represents in the area formula.