Mr. Orchard’s Math 142 WIR
Test 3 Review
Week 12
1. A runner’s speed (measured in feet per second) was measured at 2-second intervals are
given in the table below. The runner steadily increases her speed the entire time.
t
v(t)
0
0
2
2.2
4
3.2
6
4.5
8
5.8
10
6.2
12
7.1
(a) Give an upper bound for the distance traveled by the runner during this time using
n = 6.
(b) Give a lower bound for the distance traveled by the runner during this time using
n = 6.
2. Evaluate the following functions at the given point.
(a) f (x, y) = 0.9x + 0.5y − 0.05xy at (0, −1)
(b) g(x, y) = ln(x + y − 7) at (e, 7)
(c) W (a, b, c, d) =
a(1+b)−d2
2c
at (0.7, 8, 1, −7)
Mr. Orchard’s Math 142 WIR
Test 3 Review
Week 12
√
and the total
3. The marginal cost function for dog sweaters is given by C 0 (x) = 10 √x+50
x
cost of making 16 dog sweaters is $760. What is the total cost of making 64 dog sweaters?
4. Find the domain of the following functions of two variables.
√
(a) f (x, y) = 5x + y
(b) g(x, y) = ln(x − y − 10)
Mr. Orchard’s Math 142 WIR
5. Find the exact value of
region.
Test 3 Review
Week 12
√
−
4 − x2 dx by finding the area of an appropriate geometric
−2
R2
6. The graph of g(x) is given. Find the exact value of
R0
−3
g(x).
3
2
1
0
-1
-2
-3
-3
-2
-1
0
1
2
3
7. What value of B makes the average value of f (x) = 3x2 − 3x on the interval [−B, B]
equal to 9?
Mr. Orchard’s Math 142 WIR
Test 3 Review
Week 12
8. A manufacturer makes gadgets and widgets. The weekly demand and cost functions for
the consoles are
p = 300 − 2x + 2y
q = 225 − x + 6y
C(x, y) = 900 + 90x + 120y
where x represents the weekly demand for gadgets; y represents the weekly demand for
widgets; p and q represent the price (in dollars) of gadgets and widgets (respectively);
and C(x, y) is the cost function.
(a) Find R(x, y), the weekly revenue function.
(b) Find P (x, y), the weekly profit function.
(c) What is the profit earned when the manufacturer makes 4 gadgets and 8 widgets?
Mr. Orchard’s Math 142 WIR
Test 3 Review
9. The supply and demand functions for a commodity are given by
D(x) = 53.2 − 1.2x
P (x) = 0.2x2 + 10
(a) What is the market equilibrium point?
(b) What is the consumers’ surplus?
(c) What is the producers’ surplus?
Week 12
Mr. Orchard’s Math 142 WIR
Test 3 Review
Week 12
10. Find the exact values of the following integrals using The Fundamental Theorem of
Calculus.
R1
(a) −1 x3 dx
(b)
R2
(c)
R1
1
x(3x −
2
x
+ x−4 )dx
3x
dx
0 (x2 +4)2
11. Evaluate
RB
0
(x2 +
3
x+3
+ 1)dx. Assume B is a real number and B > 0.
Mr. Orchard’s Math 142 WIR
Test 3 Review
Week 12
12. Find all first-order partial derivatives of the following functions.
(a) f (x, y) = x2 + 2x3 y 4 − 13 x3 − y
(b) h(x, y, z) =
y
xz
13. If f (3) = 15, f 0 (x) is continuous, and
R3
−4
f 0 (x)dx = 15, what is the value of f (−4)?
Mr. Orchard’s Math 142 WIR
Test 3 Review
14. Use u-substitution to find the following antiderivatives.
R
(a) 2(3x − 5)2 dx
(b)
R
x2
dx
(x3 +8)3
(c)
R
6x2 +4e2x
dx
x3 +e2x
Week 12
Mr. Orchard’s Math 142 WIR
Test 3 Review
Week 12
15. Speedometer readings (in feet per second) for a motorcylce at 12-second intervals are
given in the table below:
t
v(t)
0
36
12
38
24
22
36
22
48
24
60
33
(a) Estimate the distance traveled by the motorcycle during this time using a left hand
sum with n = 5.
(b) Estimate the distance traveled by the motorcycle during this time using a right
hand sum with n = 5.
16. Find the derivative of
R3
ln(x)
2
2x dx
Mr. Orchard’s Math 142 WIR
Test 3 Review
Week 12
17. Find the average value of f (x) = x(x − 1) on the interval [−1, 2].
18. Find all the second partial derivatives of the function f (x, y) = ex
3 +y 5
Mr. Orchard’s Math 142 WIR
Test 3 Review
Week 12
19. Find the area below the x-axis and above y = (x − 2)2 − 4.
20. Use a left sum and a right sum with 4 rectangles of equal width to approximate
R1
−1
2
e−x dx