Calc H 2012 Final Study Guide
Name______Period______
Find the location of the indicated absolute extremum for the function.
2) Minimum
2) ______
Find the extreme values of the function on the interval and where they occur. Identify any critical points that are not stationary points.
4) f(x) = , - 8 ≤ x ≤ 3 4) ______
Find the extreme values of the function and where they occur.
6) y = 6) ______
A) The minimum is 0 at x = 1. The maximum is 0 at x = -1.
B) The minimum is 0 at x = 0.
C) The maximum is 0 at x = 0.
D) The minimum is - at x = -1. The maximum is at x = 1.
Give an appropriate answer.
8) Find the value or values of c that satisfy = (c) for the function f(x) = x + on the interval [3, 16].
8) ______
A) 0, 4 B) - 4, 4 C) 4 D) 3, 16
Find the local extrema.
9) k(x) = 3x2 + 12x + 9 9) ______
10) g(x) = -4x2 - 40x - 98 10) ______
11) y = - 18 + 9 11) ______
12) f(x) = 12) ______
Find all possible functions with the given derivative.
13) f'(x) = 9 - 14x + 7 13) ______
14) f'(x) = 14) ______
Find the function with the given derivative whose graph passes through the point P.
17) f'(x) = 4x - 12 + 7 cos x, P(0, 9) 17) ______
Use the Concavity Test to find the intervals where the graph of the function is concave up.
18) y = -3x2 + 18x + 4 18) ______
20) y = 5x - 6 20) ______
Use the graph of f to estimate where f'' is 0, positive, and negative.
22)
22) ______
Use the Second Derivative Test to find the local extrema for the function.
23) y = 45 - 3 23) ______
24) y = + 3x - 1 24) ______
Use the given derivative of the function to find the local extrema of the function.
25) y' = (x + 1)(x + 3) 25) ______
Sketch a graph of a single function that has these properties.
28) Using the following properties of a twice-differentiable function y = f(x), select a possible graph of f.
28) ______
Solve the problem analytically.
30) Of all numbers whose difference is 4, find the two that have the minimum product. 30) ______
Solve the problem.
31) A carpenter is building a rectangular room with a fixed perimeter of What are the dimensions of the largest room that can be built? What is its area? 31) ______
32) Find the dimensions that produce the maximum floor area for a one-story house that is rectangular in shape and has a perimeter of 159 ft. Round to the nearest hundredth, if necessary. 32) ______
Find the linearization L(x) of f(x) at x = a.
34) f(x) = x + , a = 3 34) ______
Solve.
36) Find dy given y = 8 + 5x - 4. 36) ______
Find the differential.
38) d(csc(6 - 1)) 38) ______
Solve the problem.
39) The radius of a right circular cylinder is increasing at the rate of , while the height is decreasing at the rate of . At what rate is the volume of the cylinder changing when the radius is 17 in. and the height is 5 in.?
39) ______
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.
42) f(x) = , a = 1, b = 5
Use MRAM with two rectangles of equal width. 42) ______
A) B) C) D)
Use a calculator or computer program to solve the problem.
43) Use RRAM to estimate the area of the region enclosed between the graph of and the x-axis for 0 ≤ x ≤ 2; n = 50. 43) ______
44) Use RAM to estimate the area of the region enclosed between the graph of and the x-axis for 0 ≤ x ≤ 4 44) ______
45) Use RAM to estimate the area of the region enclosed between the graph of and the x-axis for 1 ≤ x ≤ 8 45) ______
Express the limit as a definite integral.
47) △, [3, 5] 47) ______
Use NINT on a calculator to find the numerical integral of the function over the specified interval.
49) y = 6tan x ; from x = 0 to x = 49) ______
USE NINT to find the average value of the function on the interval. At what point in the interval does the function assume its average value?
50) y = , [0, 3.87298335] 50) ______
51) y = - 6 - 1, [0, 3.46410162] 51) ______
Find the average value of the function without integrating, by appealing to the geometry region between the graph and the x-axis.
53) f(t) = 2 - , [-2, 2]
53) ______
Find the average value over the given interval.
54) y = ; [1, e] 54) ______
55) y = 10 sin x; [0, π] 55) ______
Find dy/dx.
56) 56) ______
57) 57) ______
58) 58) ______
Evaluate the integral.
59) dx 59) ______
60) dx 60) ______
Find the total area of the region between the curve and the x-axis.
61) y = 2x - ; 0 ≤ x ≤ 2 61) ______
Use NINT to solve the problem.
64) Evaluate . 64) ______
A) 0.05048 B) 0.88623 C) 0.00248 D) 0.00415
65) Evaluate dx. 65) ______
Solve the problem.
68) A rectangular swimming pool is being constructed, 18 feet long and 100 feet wide. The depth of the pool is measured at 3-foot intervals across the width of the pool. Estimate the volume of water in the pool using the Trapezoidal Rule.
68) ______
Use the Trapezoidal Rule to estimate the integral.
69) , n = 4 69) ______
Evaluate the integral.
71) dt 71) ______
Evaluate the integral using the given substitution.
73) dt, u = 1 - sin 73) ______
74) , u = 8x + 12 74) ______
Evaluate the integral.
76) dx 76) ______
77) 77) ______
78) 78) ______
Evaluate the definite integral by making a u-substitution and integrating from u(a) to u(b).
79) dx 79) ______
Use tabular integration to find the antiderivative.
82) 82) ______
Find the area of the shaded region.
83) f(x) = + - 6x
83) ______
84) f(x) = - + + 16x
84) ______
85)
y = - 2x 85) ______
86) y = 2 + x - 6 y = - 4
86) ______
Find the area of the regions enclosed by the lines and curves.
88) y = 9x - and y = 20 88) ______
89) About the x-axis
2) ______
A) 8π B) 4π C) 16π D) 14π
90) About the y-axis
3) ______
A) π B) π C) π D) π
Theorem:
1. MVT
2. FCT Part I and II
Key:
2) x = -2
4) Maximum value is at x = - 8; minimum value is at x = 3
6) The minimum is - at x = -1. The maximum is at x = 1.
8) 4
9) Local minimum at (-2, -3)
10) Local maximum at (-5, 2)
11) Local minima at (3, -72), (- 3, -72); local maximum at (0, 9)
12) No local extrema
13) 3 - 7 + 7x + C
14) 168 ln x + C
17) 2 - 12x + 7 sin x + 9
18) None
20) None
22) Zero: x = 0; positive: (0, ∞); negative: (-∞, 0)
23) Local minimum: (- 3, - 486),local maximum: (3, 486)
24) Local minimum:
25) Local maximum at x = -3; local minimum at x = -1
28)
30) 2 and - 2
31) 125 ft × 125 ft; 15,625
32) 39.75 ft × 39.75 ft
34) L(x) = x +
36) (16x + 5) dx
38) - 12x csc(6 - 1) cot(6 - 1) dx
39) 323π in.3/s
42)
43) 1.336
44) 25.333
45) 2.0794
47) C) dx
49) 0.86304469
50) - , at x = 2.23606798
51) -25, at x = 2
53)
54) C)
55)
56) 8
57)
58)
59) 15 - ln 16
60) 2 + 4π
61) C)
64) 0.00415
65) ≈ 1.133
68) 12,300
69) - π
71) + 7 cos t + C
73) - + C
74) + C
76) + C
77) - cos (9x - 8) + C
78) - cot (10θ + 5) + C
79) C) - 18
82) [ - 7x + 7] + C
83)
84)
85)
86)
88) C)
89) C) 16π
90）C) π
11