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1. If f(x) = , then f ‘ (x) =
2.
391
13
1
33
2
3
1
3 222
2
32
2
2
x
x
x
x
x
xx )e(
)(
)( )d(
)( )c( )b( )a(
1
32 x
x
Advanced Placement Calculus Semester One Review Name:___________________Calculator Active Multiple Choice
h
xxhh
tan)tan(lim
0
sec (e) sectan )d( sec )c( cot )b( cot )a( 22 xxxxxx
(e) )d( )c( )b( )a(2
2
2
22
2
2
2
2
21222 y
x
y
yx
yx
yx
yx
yx
yx
yx
dx
dy3. If x3 + 3xy + 2y3 = 17, then
4. The derivative of f is x2(x - 2)(x + 3) . At how many points will
the graph of f have a relative maximum?
Four (e) Three )d( Two )c( One )b( None )a(
5. If f (x) = sec x + csc x, then f ‘(x) =
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xxxx
cotcsctansec)e( cotcsctansec)d(
seccsc)c( cscsec)b( )a(
220
)arctan()( 2xxxf
. and . (e) . and , , . )d(
. )c( 0.294 )b( . )a(
29405481316103161
29405481
xxx
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)(
(e) )('
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1
222
The circumference of a circle is increasing at a constant rate of 5 feet per second. Find the rate at which the radius is increasing.
ft/sec 10 (e) ft/sec 5 )d( ft/sec )c( ft/sec )b( ft/sec )a( 105
2
5
8.
6. Find the x-coordinates of all points of inflection for the function
7. Let ( ) ( ), find '( )g x f x g x
Advanced Placement Calculus Semester One Review Name:___________________Calculator Free Response
Suppose the distance in feet covered by a car moving along a straight road t seconds after starting from rest is given by the function f (t) = 3.12t2 sin(0.3t)
a. Find the average velocity of the car over the interval for t = 13 to t = 33 seconds.
b. Find the time t in the interval t = 13 to t = 33 where the velocity of the car is equal to the average velocity.
c. Find the instantaneous velocity when t = 33 seconds
d. Find the acceleration of the car when t = 33 seconds.
Let A(x) be the area of the rectangle bounded by the x- and y- axis and the curve
as shown below.
A. Find the dimension of this rectangle such that its area is a maximum.
B. Verify that when the area is a maximum the length of the rectangle is the same as x-coordinate of the point of inflection of the graph of y.
2xey
2xey
c
x
y
1. If , then f ‘ (8)=
2.
1 (e) )d( )c( )b( )a(4
5
11
3
5
40
3
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3
40
3
410 )e( )d( )c( )b( )a(
524
1352
2
xx
xxx
lim
34
5)( xxf
Advanced Placement Calculus Semester One Review Name:___________________No Calculator Multiple Choice
5. Find the limit .
(a) -1 (b) 0 (c) 1 (d) 2 (e) This limit does not exist.
223
2
2
xxx
xlim
2 (e) 2)d( 1)c( 0)b( )a( 2
3. If f (x)=cos2x, then f ´´ ()=
4. If is both continuous and differentiable, then a =
14
162
3
xbx
xxaxxf
;
;)(
26 (e) 14 )d( 4 )c( 14- )b( )a( 26
7. What is the slope of the tangent line to the curve
at the point where x = 2?
8. For what value of x does the function have a relative minimum?
61209 23 xxxxf )(
12
2
x
xy ln
(e) )d( )c( )b( )a(9
2
5
1
7
4
5
31
minimum. relative no has function This (e) 2- )d( 3 )c( 4 )b( )a( 10
6. In the given interval 0 < x < 2, the function has critical numbers atxxy21sin
34
32 , (a) x
35 , )b(
3x
611 , (c)
6x
67, (d)
65x
these of None (e)
9. The equation of the line tangent to at the point (1,5) is23
32
x
xy
(e) )d(
)c( )b( 13 )a(
yx-yx
yxyxyx
13326613
641318138
-7 -6 -5 -4 -3 -2 1 0 1 2 3 4 5 6 5
The graph of the derivative of f (x) or f ’ (x) is shown. f ’ (x) has horizontal tangents at the points x = -5, -2, and 2.
a. Find the x-coordinate(s) of any relative extrema of f (x) and classify each as maximum or minimum. Justify your answer(s).
b. Find the interval(s) where f (x) is concave down.Justify your answer(s)
c. Find the x-coordinate of the absolute maximum of f (x).
Advanced Placement Calculus Semester One Review Name:___________________No Calculator Free Response
Graph of y = f´(x)
A circle is inscribed in a square as shown to the right. The circumference of the circle is increasing at a constant rate of 5 feet per second. As the circle expand, the square expands to maintain the condition of tangency. (Hint: A circle with a radius of r had a circumference of C=2r and an area of A = r 2)
A. Find the rate at which the perimeter of the square is increasing. Be sure to indicate units of measure.
B. At the instant when the area of the circle is 25 in2, find the rate of increase in the area enclosed between the square and the circle.