Ap Calc Final Review Sheet

Front 1

Find the zeros of a function.

Back 1

Set the function equal to zero and solve for x.

Front 2

Find equation of the line tangent to f(x) at
(a,b).

Back 2

Find the derivative of the function, write equation in point slope form.

Front 3

Find equation for the line normal to f(x) at (a,b).

Back 3

Find the slope of the tangent line, then find the negative reciprocal of that slope. Then point slope form.

Front 4

Show that f(x) is even.

Back 4

f(-x) = f(x)

Front 5

Show that f(x) is odd.

Back 5

f(-x) = -f(x)

Front 6

Find the interval where f(x) is increasing.

Back 6

Find f'(x). Set both the numerator and denominator to zero to find the C.P.'s. Make a sign chart of f'(x) and determine where f'(x) is positive.

Front 7

Find the interval where the slope of f(x) is increasing.

Back 7

Find f''(x). Set both the numerator and denominator to zero to find the C.P.'s. Make a sign chart of f''(x) and determine where it is positive.

Front 8

Find the minimum value of a function.

Back 8

Make a sign chart of f'(x). Find all relative minimums and plug those values back into f(x). Choose the smallest.

Front 9

Find the minimum slope of a function.

Back 9

Make a sign chart of the derivative of f'(x) = f"(x). Find all relative minimum and plug those values back into f'(x) and choose the smallest.

Front 10

Find critical values.

Back 10

Set f'(x) equal to zero. With a rational expression, set the denominator to zero as well.

Front 11

Find inflection points.

Back 11

Set f"(x) equal to zero to find P.I.P.'s. Make a sign chart with P.I.P.'s and f"(x) to find where concavity changes.

Front 12

Show that lim f(x) exists
x->a

Back 12

lim f(x) = lim f(x)
x->a- x->a+

Front 13

Show that f(x) is continuous.

Back 13

Show that:
1. lim f(x) exists
x->a
2. f(a) exists
3. lim f(x) = f(a)
x->a

Front 14

Find vertical asymptotes of f(x).

Back 14

Factor and cancel as much as possible. Set the denominator to zero.

Front 15

Find horizontal asymptotes of f(x).

Back 15

Find lim f(x)
x->*

Front 16

Find average rate of change of f(x) on [a,b]

Back 16

f(b) - f(a) / b - a

Front 17

Find the instantaneous rate of change of f(x) at a.

Back 17

Find f'(x), plug in and solve for a.

Front 18

Find the average value of f(x) on [a,b]

Back 18

1/b-a a∫b f(x)dx

Front 19

Find the absolute maximum of f(x) on [a,b]

Back 19

Make a sign chart of f'(x), find all relative maximums and plug values back into f'(x) as well as end points. Find largest value.

Front 20

Show that a piecewise function is differentiable at the point a where the function rule splits.

Back 20

lim f(x) = lim f(x)
x->a- x->a+

Front 21

Given s(t) (position function), find v(t).

Back 21

s'(t) = v(t)

Front 22

Given v(t), find how far a particle travels on [a,b].

Back 22

Integral from a to b for absolute value of v(t).

Front 23

Find the average velocity of a particle on [a,b]

Back 23

one over b - a etc = f(b) - f(a)/b -a

Front 24

Given v(t), determine if a particle is speeding up at t = k.

Back 24

Find v(k) and a(k). If the signs are the same, the particle is speeding up.

Front 25

Given v(t) and s(0), find s(t).

Back 25

s(t) + C = ∫v(t)dt

plug in t = 0 to find C, then particular solution

Front 26

Show that Rolle's Theorem holds on [a,b].

Back 26

Show that the function is continuous and differentiable on the interval. If f(a) = f(b), then find some c in [a,b] such that f'(c) = 0

Front 27

Show that the Mean Value Theorem holds on [a,b]

Back 27

Show that the function is continuous and differentiable on the interval.
f'(c) = f(b) - f(a) / b - a

Front 28

Find domain of f(x).

Back 28

Assume domain is (-*,*)
Restrictions: denominators going to zero, square roots of negatives, log/ln of negative, and some trig functions (sec, csc, tan, cot)

Front 29

Find range of f(x) on [a,b]

Back 29

use max/min techniques to find relative max/mins then examine f(a) and f(b)

Front 30

Find range of f(x) on (-*,*)

Back 30

use max/min techniques to find relative max/mins, then examine lim f(x)
x->+-*

Front 31

Find f'(x) by definition

Back 31

f'(x) = lim f(x+h) - f(x) / h defined function
h->0
f'(x) = lim f(x) - f(a) / x - a defined value
x-> a

Front 32

Find derivative of inverse to f(x) at x=a

Back 32

If g(a) = f-1(a) g'(a) = 1/ f'(g(a))

Front 33

y is increasing proportionally to y

Back 33

dy/dt = ky ----> y=PE^rt

Front 34

Find the line x = c that divides the area under f(x) on [a,b] to two equal area

Back 34

two integrals equal to one another, one from a to c, one from c to b

Front 35

d/dx a∫x f(t)dt =

Back 35

FTC part 2. Plug in x. f(x)

Front 36

d/dx a∫u f(x)dt

Back 36

2nd FTC f(u)du/dx

Front 37

The rate of change of population is...

Back 37

dP/dt = ...

Front 38

The line y=mx + b is tangent to f(x) at (a,b)

Back 38

Show that f'(a) = m and that the points (a, b) are on both the function f(x) and the line

Front 39

Find area using left Riemann sums

Back 39

A = base[f(x0)+f(x1)+...f(xn-1)]

Front 40

Find area using right Riemann sums

Back 40

A = base[f(x1)+f(x2)+...f(n)]

Front 41

Find area using midpoint rectangles

Back 41

A = base[f(x1)+f(x3)+...f(fn-1)]

Front 42

Find area using trapezoids

Back 42

A = 1/2(base)[f(x0)+2f(x1)+...f(xn)]

Front 43

Solve the differential equation ...

Back 43

Separate the variables, integrate both sides

Front 44

Meaning of a∫x f(t)dt

Back 44

the accumulated area under the function f(t) starting at some constant a and ending at x

Front 45

Given a base, cross sections perpendicular to the x-axis are squares

Back 45

a∫h (base)^2dx a∫b (f(x)-g(x))^2

Front 46

Find where the tangent line to f(x) is horizontal

Back 46

set f'(x) equal to zero and solve (only the numerator)

Front 47

Find where the tangent line to f(x) is vertical

Back 47

set f'(x) equal to zero and solve (only the denominator)

Front 48

Find the minimum acceleration given v(t)

Back 48

Find the acceleration a(t) = v'(t). Then minimize the acceleration by examining a'(t)

Front 49

Approximate the value of f(0.1) by using the tangent line to f at x = 0

Back 49

Find the equation of the tangent line of f using y-y1 = m(x-x1) where m = f'(0) and the point is (0,f(0)). Then plug in 0.1 into the line, using ~ sign

Front 50

Given the value of f(a) and the fact that the anti-derivative of f is F, find F(b)

Back 50

a∫b f(x)dx = F(b) - F(a)

Front 51

Find the derivative of f(g(x))

Back 51

chain rule. f'(g(x))g'(x)

Front 52

Given a∫b f(x)dx, find a∫b[f(x)+k]dx

Back 52

a∫b f(x)dx + a∫b kdx

Front 53

Given a picture of f'(x), find where f(x) is increasing

Back 53

where the graph is above the x-axis

Front 54

Given v(t) and s(0), find the greatest distance from the origin of a particle on [a,b]

Back 54

s(0) + a∫v(t)=0 v(t)dt

Front 55

Given a chart of x and f(x) on selected values between a and b, estimate f'(c) where c is between a and b

Back 55

pick a value k less than c and a value h greater than c
f'(c) ~ f(k) - f(h) / k - h

Front 56

Given dy/dx, draw a slope field

Back 56

use the given points and plug them into dy/dx, drawing lines w/ indicated slopes at the points

Front 57

Find the area between the curves f(x), g(x) on [a,b]

Back 57

a∫b [f(x)-gx)]dx

Front 58

Find the volume if the area between f(x), g(x) is rotated about the x-axis

Back 58

V = pi a∫b [(f(x))^2 - (g(x))^2]dx