Ap Calc Review

Limits

100

What law is used when a limit equals 0/0?

L'Hopital's Rule.

200

Find  

lim_(x to 0) (9 sin)/(3 x)

What is: 27

300

What does the squeeze theorem state?

If f(x)≤g(x)≤h(x), and the limits of f(x) and h(x) as x approaches a point a are both equal to the same value L, then the limit of g(x) as x approaches a much also be L

400

Find

lim_(x to -1) (x+1)/(x^3+1)

What is:

1/3

500

Find the value k that makes the function continuous.

f(x) = {(kx-90 text{ if } x>=8), (-6x-10 text{ if } x<8)}

What is: k = 4

Derivatives

100

d/dx ((x²+1)(x-4))

2x(x−4)+(x²+1)

200

Find the derivative of the following function: 

y=e^{(-5x) ^4}

y'=e^{(-5x)^4} *(-20x³)

300

find the derivative of the following function: 

y=ln(-5x³+9x²)

y'=1/(-5x³+9x²)*(-15x²+18x)

400

Given f(x)=−4x²+2x+5, find the slope of the tangent line of f at the point where x=−3

26

500

If f(6) = −2 and f ′(6) = 1/3 , what is the derivative of 

y = [ f(2x)]^3 at x = 3?

8

In Motion

100

How do you find velocity given position?

Take the derivative.

200

Position of a particle at x=0 is given by the equation 5x²+8. What is the particles velocity at t=0

10x

300

A particle moves along a line with position

s(t)=t³−6t²+9t 

At what time(s) is the particle at rest?

t= 1, 3

400

A particle moves along the x-axis so that at time t≥0 its position is given by x(t)=t3+5t2-32t. Determine the velocity of the particle at t=1.

-19

500

A particle moves along the x-axis so that at time t≥0t≥0 its velocity is given by v(t)=16t³−48t. Determine all intervals when the acceleration of the particle is positive.

(1, infinity)

High/Lows

100

If f'(x)>0, then f(x) is 

(a) Constant 

(b) Increasing 

(c) Decreasing 

(d) Concave down

(c) Increasing.

200

If f′(2)=0 and f′′(2)>0, then x=2 is:

(A) A point of inflection
(B) A local minimum
(C) A local maximum
(D) Neither a max nor min

(B) A Local minimum

300

The slope of the normal line to the curve y= (2x²) + 1 at (1,3) is

What is -1/4 ?

400

The x-value where the function f given by 

f(x) = x⁴ +1−16ln x has a relative minimum

x = sqrt 2

500

Given the function f(x)=x4−4x3,f(x)=x4−4x3, determine all intervals on which f′f′ is decreasing.

 

[0,2]

Inverse Trig

100

What is the inverse trig function of sin(x)?

(1)/(1-x²)^1/2.

200

What is the inverse trig function of cos(x)?

-(1)/(1-x²)^1/2.

300

What is the inverse trig function of tan(x)?

(1)/(1+x²).

400

What is the inverse trig function of sec(x)

(1)/(x²(1-1/x²)^1/2).

500

What is the inverse trig function of cot(x)?

-(1)/(1+x²)