AP AB Calculus Unit 2 Review

Theorem: Differentiable Implies Converse If f is differentiable at x=c then f is continuous at x=c True or False: The converse to the theorem above is also true. If the converse is not Tue, sketch a function to the right that serves as a counter-example

False: there's a sharp turn/cusp

Derivative of each of the following: a. y=2x^7 b.g(x)=3/x^2 c. f(x)= 6^√x^5 /8 d. y=3x^4/8 e. y=(3x)^4 /8 f. y=9/7x^2 g. y=9/7x^2

a. 14x^6 b. -6/x^3 c. 5/48x^1/6 d. 3/2 *x^3 e. 81/2 *x^3 f. -18/7x^3 g. -18/49x^3

Find the derivative of each of the following: a.f(x)= x^3- 4x +5/x b.g(x)= (x^2 +1)(x-3) c. f(x)=(x-3)^3 d. g(x)= 2x^2 -3x/ √x

a. 2x-5/x^2 b. 3x^2 -6x +1 c. 3x^2 -18x +27 d. 3x^1/2 -3/2x^1/2

One lazy day, Uncle Si decides to sit under a shade tree next to a road by the swamp and count the nuber of ducks which cross the road. The data in the table below shaws the accumulation of the number of ducks crossing this road at each hour after 9:00 am. Hours after 9 am: 0 1 2 3 4 5 6 7 # of ducks that crossed: 0 3 8 11 12 21 24 28 a. Determine the average number of ducks which have crossed the road per hour during Uncle Si's 7-hour observation. Label your results. b. Estimate the value of f'(4.5) and explain its meaning.

a. 4 ducks/ hr b. 9 ducks/hr were passing through at around 1:30

Suppose that f(x) and g(x) and their derivatives have the following values at x=-1 and x=0. x: 0 -1 f(x): -1 0 g(x): -3 -1 f'(x): -2 2 g'(x): 4 1 a. 3*term-13f(x) -g(x) x=-1 b. 3*f(x) * g(x) x=-1 c. (f(x))/ g(x) +2

a. 5 b. -6 c. 6

Find the derivative of each of the following: a.f(x)=x^5 b.g(x)=4^√x^3 c. y=1/x^3

a. 5x^4 b. 3/4x^1/4 c. -3/x^4

Find f and c. a. lim △x->0 [5-3(1+△x]-2/ △x b. lim △x->0 ((-2+△x)^3 +8/△x

a. c=1 f(x)=5-3x b. c=-2 f(x)=x^3

Compute f'(c) for each of the following using the alternate form of the definition of the derivative. a. f(x)=3x+1, c=1 b. f(x)= x+ 4/x, c=4

a. f'(x)=3 b.f'(x)=3/4

Compute the Derivative function, f'(x) using the definition of derivative for each of the following. a. f(x)=3x^2 +1 b. f(x)=2 c. f(x)= 3/x+1 d. f(x)= √3x+1

a. f'(x)=6x b. f'(x)=0 c. f'(x)= -3/(x+1)^2 d. f'(x)= 3/2√3x+1

The limits below represent f'(c) for a function f and a number c. Find f and c. a. lim h->0 √(9+h) -3/ h b. lim △x-> [2+(-3+△x)^3]- (-25)/ △x c. lim △x->0 ((x+△x)^3 -5(x+△x)^2 +4(x+△x) -7 -(x^3 -5x^2 +4x -7))/△x

a. f(x)= √x c=9 b. f(x)= x^3 +2 c=-3 c. f(x)= x^3 =5x^2 +4x -7 c=x

For the function f(t)= (t^3 +2)/t, find the following. a. the average rate of change of f(t) on the interval [1, 4] b. the instantaneous rate of change of f(t) when t=2.

a. m=9/2 b. f'(2)= 7/2

let f(x)={ ax, x≤1 } .Find all possible values of a and b bx^2 +x +1, x>1 such that f(x) is differentiable at x=1. Show proper justification.

a=3 b=1

Let f(x)= √x+2 . Which of the following gives the correct limit definition of the derivative of f(x) evaluated at x=6? a. f'(6)= √x+2 -2√2/ x-6 b. f'(6)= limx->2 √x+2 -2√2/ x-2 c. f'(6)= lim x->6 √x+2 -2√2/ x-6 d. f'(6)= lim x->6 √x+2 -√6/ x-6

c.

If f(x)= 2x^2 +4 , which of the following will calculate the derivative of f(x)? a. ([2(x+△x)^2 +4] - (2x^2 +4))/ △x b. lim △x->0 ((2x^2 +4 +△x) - (2x^2 +4))/ △x c. lim △x->0 ([2(x+△x)^2 +4] - (2x^2 +4))/ △x d. ((2x^2 +4 +△x) - (2x^2 +4))/△x

c. lim △x->0 ([2(x+△x)^2 +4] - (2x^2 +4))/ △x

Graphs with a vertical tangent Lines: Derivative will not exist Ex. Sketch the graph and find the derivative of f(x)=x^1/3 when x=0.

f'(0) Does Not Exist *see graph in notes*

Graphs with Sharp Turns: derivative will not exist Ex. Sketch the graph and find the derivative of f(x)= |x-2| when x=2.

f'(2) Does Not Exist *see graph in Notes*

If the derivative of f'(4) lies between the points (0,1) and (6,13) what is f'(4).

f'(4)=2

Theorem: Equation of a Tangent Line: y-f(c)=f'(c)(x-c)

f'(c)=slope f(c)=y-variable c=x variable

Differentiate f(x)= √x and determine the domain of f'(x)

f'(x)= lim h->0 1/2√x

lim △x->0 ([2{-2+△x)^3 -2(-2 +△x) -2] -(-14))/ △x represents f'(c) for a function f(x) and a number c. Find f(x) and c.

f(x)= 2x^3 -2x -2 c=-2

Suppose that h(x)= (g(x))/(f(x)) g(2)=3 g'(2)=-1 f(2)=5 f'(2)=-2 FIND h'(2)

h'(2)= 1/25

Consider the limit lim x->c (f(x)-f(c))/x-c. Let x=c+h Rewrite and simplify, the limit above using the new expression for x.

lim h->0 (f(c+h) - f(c))/ h

Differentiate: y=(3x)/(x^2 +1)

y'=(-3x^2 +3)/(x^2 +1)^2

See example problem #7 for graph in order to complete... Notes 2.4 The function f is defined on the closed interval [-2,16]. The graph of the derivative of f, y=f'(x), is given below. The point (14,-2) i on the graph y=f(x). An equation of the tangent line to the graph of f at (14,-2) is...

y+2=3(x-14)

If f(2)=3 and f'(2)=-1, find the equation of the tangent line when x=2.

y=-x+5

Find the equation of the tangent line to the graph of f(x)=x^2 -2x -3 when x=2.

y=2x-7

The derivative of Function, f' f'(x)= lim h->0 (f(x+h) -f(x))/ h or f'(x)= lim△x->0 (f(x+△x) -f(x))/ △x

△x means change in x

If the line 2x+3y=k is tangent to the graph of y=f(x) at the point where x=5, the value of lim x->5 (f(x)-f(5))/x-5 is....

-2/3

*See Unit 2 Review part 1 for the graph of questions 10 and 11 10. Let h(x)= f(x)*g(x) Find h'(2) 11. What is g'(3)?

10. h'(2)=7 11.g'(3) is the slope of g at x=3. In the case of the graph shown the slope is undefined at g of x=3.

Find the slope of the graph of f(x)= 2x-3 at the value x=1

2

Find the slope of the tangent line to the graph of f(x)= x^2 + 1 at the value of x=1

2

If f is a function for which lim x->-2 (f(x)-f(-2))/ x+2 =0, then which of the following statements must be true? a. x=-2 is a vertical asymptote of the graph b. the function f if continuous at x=2 c. the derivative of f at x=-2 exists d. f is not defined at x=-2

C.

The limit lim △x->0 ((1+△x)^2 -(1))/ △x equals f'(c) for some function, f(x), and some constant, c. Determine f(x) and c. a.f(x)=x^2 +1, c=1 b. f(x)= x, c=-1 c. f(x)= x^2, c=1 d. f(x)= x^2, c=-1

C. f(x)= x^2 and c=1