Mr. Orchard’s Math 142 WIR
1. Find all antiderivatives of f 0 (x).
(a) f 0 (x) = x3 + 3x
(b) f 0 (x) = ex − 4x
(c) f 0 (x) = x +
1
x
√
0
(d) f (x) =
ln(x)
x
(e) f 0 (x) =
e2x −e−2x
e2x +e−2x
Test 3 Review
Week 13
Mr. Orchard’s Math 142 WIR
2. Given f 0 (x) =
−24x
(3x2 +1)2
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Week 13
and f (1) = 10, find f (x).
3. An object travels at v = 5t2 + 1 feet per second. Estimate the distance the object travels
in the first two seconds using n = 5 equal sub intervals and a right hand sum.
4. The following table gives the velocity at 1-second intervals of an accelerating car. Use
the information to give an upper and lower estimate of the distance the car travels.
0
1
2
3
4
t (s)
v(t) ft/s 10 20 25 40 50
Mr. Orchard’s Math 142 WIR
Test 3 Review
Week 13
5. Below is the graph of f 0 (x).
4
3
2
1
0
-1
-2
-3
-4
-4
(a) Find
-3
R4
−4
-2
-1
0
1
2
3
4
f 0 (x)dx
(b) Given that f (0) = 14, find f (4).
6. Approximate
R2
−2
(1 − x2 )dx using n = 4 equal sub intervals and a left hand sum.
Mr. Orchard’s Math 142 WIR
Test 3 Review
7. Solve the following definite integrals exactly.
R2
(a) −2 (1 − x2 )dx
(b)
R3
(c)
R2
(d)
RT
(e)
R0
3
2
e−x dx
−1
|x|3 dx
4t
dt
0 t2 +1
−1
2 +2
4xex
dx
Week 13
Mr. Orchard’s Math 142 WIR
Test 3 Review
8. Use the Fundamental Theorem of Calculus to find a function whose derivative is
9. Write the
R 10
5
f (x)dx −
R8
5
f (x)dx +
R2
2
f (x)dx as a single definite integral.
10. Find the average values of the following functions on the given interval.
(a) f (x) = x2 − 2x on [0, 10]
(b) f (x) = 100e−0.2x on [10, 20]
Week 13
√
9 − x2
Mr. Orchard’s Math 142 WIR
Test 3 Review
Week 13
11. Find the area between y = 2x2 + 4 and y = 3x on the interval [1, 2]
12. Find the area bounded by y=1 − x4 and y = 5x − 5.
13. Find the area between the functions y = x2 − 1 and y = 1 − x on the interval [0, 2].
Mr. Orchard’s Math 142 WIR
Test 3 Review
Week 13
14. Below are given supply and demand curves for a commodity. Find the consumer surplus
and the producer surplus.
D(x) =
S(x) =
15. Find the domains of the following functions.
(a) f (x, y) = ln(4x + 2y) +
(b) g(x, y) =
√
x+y−
2
xy
1
x(x+1)
64 − x2
3x2
Mr. Orchard’s Math 142 WIR
16. f (x, y) =
√1 ,
x−y
and g(x, y) =
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2y
.
x+y
Find the following quantities if they exist.
(a) Calculate f (1, 2)
(b) Calculate g(3, 3)
(c) Calculate f (3, 2) − g(2, 1)
17. Find all first order partial derivatives of f (x, y, z) = xy ln(y + 2z).
Week 13
Mr. Orchard’s Math 142 WIR
Test 3 Review
18. Find the following partial derivatives at the given points.
(a) fxx (1, 1) if f (x, y) = (x2 − y 3 )6
(b) fxy (1, 4) if f (x, y) = 4x3 y 2 − x2 + 2y 2 .
(c) fyy (2, 0) if f (x, y) = x9 y 7 + 3xy 5 + x4 + 25y 2
(d) fxz (1, 0, 0) if f (x, y, z) = ln(x3 + 9yz + 2z 2 )
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