Download AP CALCULUS REVIEW FOR TEST #3

AP CALCULUS REVIEW FOR TEST #3
** Use the Power Rule for derivatives to differentiate:
1.
G ( x)  3 x 5  5 x 2  1
3.
P( x)  7 x10  5 x 6 
5.
S ( x) 
y
7.
5
x
2
3
3 6
x  
7
3
x5
 6x
5
6
3 4 5 2 9
x  x  x 
4
2
4
2.
y
4.
f ( x)  6 sin x  3 tan x
6.
y  x  3 x2 
8.
F ( x) 
5
7
x
2
3
5
x3
35 x
** Take the derivative using the either the Product or Quotient Rule.
9.
F ( x)  3x 5x
11.
y  x 3 csc x
4
3
 2x
2
 1
10.
P( x) 
3x 2
5x 3
12.
f ( x) 
4x  1
3 cos x
** Use the limit definition of the derivative at a point to find the slope at the given points:
13. f ( x)  3x 2  4 x  5 ; x  4
14.
h( x)  3x  1  2 ; at x  5
** For the following two functions…
a.
Find f ' ( x) by using the limit definition of a derivative
b.
Confirm your derivative from part (a) by using the rules for derivatives to find f ' ( x)
c.
Find the slope at the given x-value using the equation found in part (a) & (b)
d.
Write the equation of the tangent line at the given x value
e.
At what x-value(s) does the function has a horizontal tangent line?
f.
Write the equation for the normal line at the given point
15. f x   x 3  3x 2  1 ; at x  2
16.
f  x   2 x ; at x  4
** Find the equation of the tangent line (in slope-intercept form) for each of the following:
17.
19.
f x   x 3  6x 2 ; at f x  15
x5
f x  
; at f x   2
x3
18.
f x   3  2 x ; at f x   3
20.
f x   xx  3; at f x  13

for f ( x) 
sin x
.
2 x  cos x
21.
Find the equation for the tangent line at x 
22.
Where is the tangent line horizontal for the function g ( x)  4 x 3 
23.
Where is the tangent line horizontal for the function m( x) 
2
5 2
x  2 x  11
2
3
4 sec x
** Graph f ' ( x) for the following functions:
24.
26.
25.
Find the values of a and b so that f (x) is everywhere continuous.
6ax 3  5bx  12, if x  1

f ( x)  3x  x 5 , if x  1
12ax 4  10bx  6, if x  1

x 2  1 x  2
f x   
4 x  3 x  2
27.
Given the following function, prove it is continuous at x  2 .
28.
Use the Intermediate Value Theorem to prove that f x   x 2  3x  1 has a root. If possible, use
two continuous integers.
29.
Graph a function on graph paper satisfying the following conditions.
lim a( x)  2
lim a( x)  
x 2
lim a( x)  
lim a( x)  1
a (3)  3
lim k 2 cos k
31.
lim k 3 cos
34.
lim
37.
lim
x  
x 3
1
30.
k
4
3
33.
lim
36.
lim
39.
42.
lim a( x)  2
x 2
x 4
r 0
r  33  27
r
5 x 2  3x  1
x  2 x 2  4 x  7
1 1

lim r3 2
r  2 r  8
40.
k 0
q 9
h  6
1
k
9q
q 3
2k  37  5
k 6
lim 64 x 2  36 x  8x
x 
lim a( x)  3
x 
a (2)  2
n 2  2n  35
n 5 n 2  10n  25
32.
lim
35.
lim
38.
lim
v 0
4 v 8
41.
lim
3 1
1 



u 5u 5u
v 4
u 0
v  43
4v
v 4  16
Given the graphed function, h(x) below…
a) state all points of discontinuity AND what type of discontinuity it is
b) which points are continuous from one direction (right or left)
c) which points are un-differentiable and why