Mr. Orchard’s Math 142 WIR
Test 2 Review
1. Find the derivative of the following functions:
(a) y = 4x12 + 5x4 + 15x3 − 2x2 − 4x + 13
(b) y = x4 102x
(c) y = (8x−2 − 2x8 )(14x2 + 33x − 18)ex
(d) y =
(3x +55)2
(15x3 −22x2 +x+1)4
Week 9
Mr. Orchard’s Math 142 WIR
(e) y = e(4x
2 −12)
(f) y = (log2 (−5x8 + 15x + 4))
(g) y = ln
Test 2 Review
12x3 −22x+15
14x−12
4
Week 9
Mr. Orchard’s Math 142 WIR
Test 2 Review
Week 9
2. Find the x-coordinates of the absolute maximum and absolute minimum (if they exist)
of f (x) = −5x2 + 100x + 600 on the interval [4, 14].
3. The price demand function for a certain commodity is p = 4800 − 55x. Use the marginal
revenue function to estimate the revenue from the 13th item sold.
4. Find the horizontal asymptotes of f (x) =
8e−3x +22
.
−4e−3x −33
Mr. Orchard’s Math 142 WIR
Test 2 Review
Week 9
5. x = 59, 400 − 22p2 is the price-demand function for a particular brand of headphones.
(a) When the price is $24 per set of headphones, is the demand elastic, inelastic, or at
unit elasticity?
(b) In order for the revenue to increase, should the price be increased or decreased from
$24?
(c) At what price should the headphones be sold in order to maximize revenue?
(d) What is the maximum revenue?
Mr. Orchard’s Math 142 WIR
6. Find f 00 (x) given f (x) below.
(a) f (x) = 12x3 − 14x + 100
(b) f (x) = x2 ln(4x)
(c) f (x) = 3x3 25x
(d) f (x) = (5x4 + 4)5
Test 2 Review
Week 9
Mr. Orchard’s Math 142 WIR
Test 2 Review
7. Find the following limits, if they exist.
(a) lim −8x3 + 22x2 + 12
x→∞
−132x50 +12x2 +4
50 +80x49 +300x30 +15
44x
x→−∞
(b) lim
15x3 +22x2 −800
2
x→∞ −400x −1200x
(c) lim
14x5 −12x3 +40
3
2
x→∞ 70x −12x +1
(d) lim
(e) lim e−8x
x→∞
Week 9
Mr. Orchard’s Math 142 WIR
Test 2 Review
Week 9
8. A box with a square base and an open top must have a volume of 32,000 cm3 . Find the
dimensions of the box that minimize the amount of material used.
9. h(x) =
4x2 −16x+12
.
−3x2 +18x−27
(a) Find the horizontal asymptotes of h(x).
(b) Find the vertical asymptotes of h(x).
Mr. Orchard’s Math 142 WIR
Test 2 Review
10. f (x) = −4x4 + 32x2 − 20
(a) Use the Second Derivative Test to find the relative extrema of f (x).
(b) Use the First Derivative Test to find the relative extrema of f (x).
Week 9
Mr. Orchard’s Math 142 WIR
Test 2 Review
Week 9
11. Use the following information about f (x), f 0 (x), and f 00 (x) to graph f (x).
f (x) is continuous on all real numbers.
f 0 (0) does not exist.
0
f (x) > 0 on (−∞, 0) ∪ (2, ∞)
f 0 (x) < 0 on (0, 2)
00
f (x) > 0 on (−2, 0) ∪ (0, 3)
00
f (x) < 0 on (−∞, −2) ∪ (3, ∞)
5
4
3
2
1
0
-1
-2
-3
-4
-5
-5
0
5
Mr. Orchard’s Math 142 WIR
Test 2 Review
Week 9
12. Two positive numbers, x and y, multiply to 162. What are these numbers if they
minimize x + 2y?
13. Find the absolute maximum and minimum values of f (x) = x4 − 8x3 + 16x2 − 25 on the
interval [0, ∞).
14. Find the critical values of f (x) = 3x4 + 12x3 − 167.
Mr. Orchard’s Math 142 WIR
15. g(x) =
Test 2 Review
x2
x+18
(a) Find the x-coordinate(s) of any local minima.
(b) Find the x-coordinate(s) of any local maxima.
(c) Find the interval(s) where g(x) is increasing.
(d) Find the interval(s) where g(x) is decreasing.
Week 9
Mr. Orchard’s Math 142 WIR
16. f (x) = 43x
2 +24x
Test 2 Review
Week 9
.
(a) Find the slope of the line tangent to the graph at x = 0.
(b) Find the equation of the line tangent to the graph at x = 0.
17. A manufacturer needs a cylinder that will hold 2.3 liters of liquid. Determine the radius
of the cylinder that will minimize the amount of material used in its construction. Round
to 4 decimal places.
Mr. Orchard’s Math 142 WIR
Test 2 Review
Week 9
18. The graph of f 0 (x) is given below. The domain of f (x) is (−∞, 0) ∪ (0, ∞).
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
What are the critical points of f (x)?
19. Find the absolute maximum and minimum values of f (x) = 4x2 − 24x + 400 on the
interval (3, 8).
4
Mr. Orchard’s Math 142 WIR
Test 2 Review
Week 9
20. Below is the graph of P 00 (x). P (x) has a domain of all real numbers.
2
1.5
1
0.5
0
-0.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
(a) Find the inflection points of P (x).
(b) Find where P (x) is concave down.
(c) We are also given the critical values of P (x): x = −2, −0.5, and 1. Where does
P (x) have relative extrema?
3