Download AP Calculus Review for Test 1 Term 4

AP Calculus Review for Test 1 Term 4
Name__________________________
Test One Wednesday April 4 2012
Determine the limit by substitution.
x2 + 14x + 49
1) lim
x 4
Answer: 11
Determine the limit algebraically, if it exists.
10 - x
2) lim
x 10 10 - x
Answer: Does not exist
3)
x2 - 4
lim
x -2 x + 2
Answer: -4
Determine the limit graphically, if it exists.
4) lim f(x)
x 1+
Answer: 3
Find the indicated limit.
11x
5) lim
x
x 0Answer: -11
1
Find the limit.
6) Let
x
lim f(x) = 1 and lim g(x) = 5. Find lim [f(x) + g(x)]2.
10
x
10
x
10
Answer: 36
Find the limit, if it exists.
x2 - 4x + 17
7) lim
x3 + 9x2 + 8
x
Answer: 0
6x + 1
8) lim
13x - 7
x
Answer:
6
13
Find a value for a so that the function f(x) is continuous.
x2 + x + a, x < -4
9) f(x) =
x3,
x -4
Answer: a = -76
Find dy/dx.
10) y = 2 tan4x
Answer: 8 tan3x sec2x
11) y = 19x - x7
Answer:
19 - 7x6
2 19x - x7
Find dy/dx by implicit differentiation. If applicable, express the result in terms of x and y.
12) 2y + 9xy - 4 = 0
Answer:
-9y
2 + 9x
13) cos xy + x5 = y5
Answer:
5x4 - y sin xy
5y4 + x sin xy
2
At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is
normal to the curve, as requested.
14) x2 + y2 - 2x + 4y = 8, tangent at (4, 0)
3
Answer: y = - (x - 4)
2
15) x2 - y2 = 15, normal at (-8, 7)
Answer: 7x - 8y + 112 = 0
Find dy/dx.
16) y = x7/5
Answer:
dy 7 2/5
= x
dx 5
Find the derivative of the given function.
17) y = 3 sin-1 (5x4)
Answer:
60x3
1 - 25x8
Find dy/dx.
18) y = 7xex - 7ex
Answer: 7xex
Find the location of the absolute extremum for the function.
19)
Answer: x = 1
3
Find the extreme values of the function on the interval and where they occur.
20) f(x) = 1x - 2; -3 x 4
Answer: Maximum value is 2 at x = 4; minimum value is - 5 at x = -3
Determine whether the function satisfies the hypotheses of the Mean Value Theorem on the given
interval.
21) g(x) = x3/4 on [0, 3]
Answer: Yes
22) f(x) = x1/3 on [-2, 5]
Answer: No
Solve the problem.
23) The graphs below show the first and second derivatives of a function y = f(x). Select a possible
graph of f that passes through the point P.
f'
f''
A)
B)
4
C)
D)
Answer: B
24) Using the following properties of a twice-differentiable function y = f(x), select a possible graph
of f.
x
y
x<2
-2
10
-2 < x < 0
0
-6
0<x<2
2
-22
x>2
Derivatives
y > 0,y < 0
y = 0,y < 0
y < 0,y < 0
y < 0,y = 0
y < 0,y > 0
y = 0,y > 0
y > 0,y > 0
A)
B)
5
C)
D)
Answer: C
Use the First Derivative Test to determine the local extrema of the function, and identify any absolute
extrema.
25) y = xe2x
1
1
Answer: Absolute minimum at - , 2 2e
Use analytic methods to find those values of x for which the given function is increasing and those values
of x for which it is decreasing.
26) f(x) = x4 - 8
Answer: Increasing on (-2, 0) and (2,
), decreasing on (- , -2) and (0, 2)
Use the Concavity Test to find the intervals where the graph of the function is concave up.
4
27) y =
x
Answer: (0,
)
Find the points of inflection.
4
28) y = x3 - 12x2 + 10x + 47
3
Answer: (3, 5)
Give an appropriate answer.
29) Find the value or values of c that satisfy
f(b) - f(a)
= f (c) for the function f(x) = x2 + 3x + 3 on
b-a
the interval [-2, 1].
Answer: -
1
2
6
Use a finite approximation to estimate the area of the region enclosed between the graph of f and the
x-axis for a x b.
1
30) f(x) = , a = 2, b = 6
x
Use Left, Midpoint, and Right rectangular approximation with four rectangles of equal width.
Answer:
352
435
Express the limit as a definite integral.
n
2
(4 c k - 9ck + 16)
31) limn
k=1
xk, [-4, 3]
3
(4x2 - 9x + 16) dx
Answer:
-4
Evaluate the definite integral.
6
ex dx
32)
-4
1
Answer: e6 e4
9 sin x dx
33)
0
Answer: 18
-1
6x-4 dx
34)
-2
Answer:
7
4
Find dy/dx.
x
10t + 3 dt
35)
0
Answer:
10x + 3
7
x
36)
0
dt
2t + 6
Answer:
1
2x + 6
x10
37)
cos t dt
0
Answer: 10x9 cos (x5)
Find the general solution to the exact differential equation.
dy
= e4x - csc x cot x
38)
dx
Answer: y =
39)
1 4x
e
+ csc x + C
4
dy
= csc2x - 20x4
dx
Answer: y = -cot x - 4x5 + C
Solve the initial value problem explicitly.
dy
= 6x2 - 4x + 4; y = 15 when x = 1
40)
dx
Answer: y = 2x3 - 2x2 + 4x + 11
41)
dy
= sin (5x + ), y = 6 when x = 0
dx
Answer: y = -
1
29
cos (5x + ) +
5
5
Solve the initial value problem using the Fundamental Theorem. Your answer will contain a definite
integral.
dy
= cos(x2) and y = 2 when x = 7
42)
dx
x
cos(t2) dt + 2
Answer: y =
7
8
Evaluate the integral.
1
x5 dx
43)
x5
Answer:
44)
7
1
sin
+ 6 dt
t
t2
Answer:
45)
x6
1
+
+C
6
4x4
7
1
cos
+6 +C
7
t
t1/6 sin(t7/6 - 3) dt
Answer: -
6
cos(t7/6 - 3) + C
7
Solve the initial value problem using the Fundamental Theorem. Your answer will contain a definite
integral.
dy
= cos(x2) and y = 5 when x = 6
46)
dx
x
cos(t2) dt + 5
Answer: y =
6
Evaluate the integral.
cos(3 + 7)
d
47)
sin2(3 + 7)
Answer: -
48)
1
+C
3 sin(3 + 7)
sin t
(3 + cos t)3
Answer:
dt
1
2(3 + cos t)2
+C
9
Solve the problem.
49) A spherical balloon is inflated with helium at a rate of 90 ft3/min. How fast is the balloon's
radius increasing when the radius is 3 ft?
Answer: 2.50 ft/min
50) A man flies a kite at a height of 50 m. The wind carries the kite horizontally away from him
at a rate of 10 m/sec. How fast is the distance between the man and the kite changing when
the kite is 130 m away from him?
Answer: 9.2 m/sec
51) A ladder is slipping down a vertical wall. If the ladder is 10 ft long and the top of it is
slipping at the constant rate of 3 ft/s, how fast is the bottom of the ladder moving along the
ground when the bottom is 8 ft from the wall?
Answer: 2.3 ft/s
Find the area of the shaded region.
f(x) = x3 + x2 - 6x
g(x) = 6x
52)
Answer:
160
3
Find the area of the regions enclosed by the lines and curves.
53) y2 = x + 5 and x = y + 25
Answer:
1331
6
54) y = x4 - 10x2 + 36 and y = 3x2
Answer:
1436
15
10
55) A rectangular plot of farmland will be bounded on one side by a river and on the other three
sides by a single-strand electric fence. With 800 m of wire at your disposal, what is the
largest area you can enclose, and what are its dimensions?
Answer: 80,000 200 by 400
11