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AP Calculus Cheat Sheet

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Section 1

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1st derivative test

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Calculus Math

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Mar 1, 2020

Cards (23)

Section 1

(23 cards)

1st derivative test

Front

Tells us max/min, increasing/decreasing intervals, slope of a tangent line, velocity. (look for sign change).

Back

Washer method

Front

V = pi (integral) R^2-r^2 dx

Back

Average rate of change

Front

slope of secant line (slope between starting and ending point)

Back

Rolle's Theorem

Front

If f is continuous on [a,b] and differentiable, such that f(a) = f(b), then there exists at least one point such that f'(c) = 0. In other words, if f(a) = f(b), then there must be a max or min between a and b.

Back

Fundamental theorem of calculus

Front

Back

Total distance traveled

Front

integral |V(t)| dt speed = |Velocity| if velocity and acceleration are in the same direction (same signs), object is speeding up. If not, object is slowing down

Back

Disc Method

Front

V= pi (integral) R^2 dx

Back

Exponential growth

Front

Pe^rt, this is a function whose rate of growth is proportional to the amount present.

Back

2nd derivative test

Front

Tells us inflection points, concavity (>0 is concave up, <0 concave down), and acceleration. (look for sign change). If first derivative is zero, and function is concave down, then f has a local maximum. If first derivative is zero, and function is concave up, f has a local minimum.

Back

Definition of inflection point

Front

x value, when function changes concavity

Back

What do we do with derivatives?

Front

Set them equal to zero to find critical values and set up a sign chart. Or, find the anti derivative (integrate) to find area underneath (area under a rate of change graph equals total change).

Back

Instantaneous rate of change

Front

Slope of tangent line (derivative) If a function is differentiable at a point, then it is continuous. The converse is not always true. Just because a function is continuous does not mean it is differentiable (sharp turn).

Back

Limits

Front

when finding limits at a number, try plugging the number in first. If that doesn't work, try factoring, using common denominators, or using conjugates. Remember, left and right sided limits must be equal for limit to exist. When approaching infinity, look for the largest exponent. If it is on top, limit is infinity (or neg. infinity), if it is on the bottom, then the limit is zero, and if they are equal, then the limit is the coefficients of top over bottom.

Back

Average value of a function

Front

1/b-a (integral a-b) f(x)dx

Back

Definition of continuity

Front

lim x->c f(x) = f(c) If and only if f is continuous at x=c

Back

Definition of concavity

Front

If f'(x) is increasing, concave up (tangent line approximation will be above, or greater than actual value of function). If f'(x) is decreasing, concave down (tangent line approximation will be less than actual value).

Back

Mean value theorem

Front

if f is continuous on [a, b] and differentiable, then at some point between a and b, ( f(b)-f(a) ) / (b-a) = f'(c). In other words, the instant rate of change will equal the average rate of change at some point

Back

Area between two curves

Front

A = (integral) [ f(x) - g(x) ] dx -"higher" function goes first When integrating to the y axis, everything must be in terms of y. Example:

Back

Total displacement (change in position)

Front

integral V(t) dt

Back

Definition of critical number

Front

x value, when first derivative is zero or undefined (does not require sign change)

Back

extreme value theorem

Front

If f is continuous over a close interval, then f has a maximum and minimum value over that interval. Also, every closed endpoint is an extreme. Remember, always consider endpoints if interval is closed when looking for extremes.

Back

Total net change

Front

(integral a-b) f(x)dx= F(b)-F(a)

Back

intermediate value theorem

Front

If a function is continuous on [a, b], then it passes through every value between f(a) and f(b).

Back