Math 110 Technical Calculus – Review, Spring 2011
1. Compute each of the following:
x2 − 4
(a) lim
x→2 x − 2
1
(d) lim √
a→−∞
1 − 2a
√
x + 25 − 5
(b) lim
x→0
x
−b2
(e) lim −e
b→∞
(c)
lim ln x
1
(f) lim+
x→1 (x − 1)
2. Using the definition of derivatives, find f 0 (x):
√
(b) f (x) = x + 2
(c) f (x) = x2 − 2x
(a) f (x) = x1
3. Find
dy
dx
x→∞
(d) f (x) = 4x2
for the following functions (some are implicit)
(a) x2 + xy 2 = 2011
(c) y = (10x + x1 + 1)3
x
(e) y = ex
(b) x2 + y 3 = 3x2 y
(d) f (x) = x2 ln x at x = e
(f) f (x) = ln(3x2 + 2x + 3)
4. A spherical balloon is increasing at the rate of 20 cubic inches per minute. At what rate
is the radius increasing when the radius is 3 inches? (Hints: V = 43 πr3 ).
5. A ladder 10 feet tall is leaning against the wall. Somebody pulled the ladder away
horizontally at a rate of 4 feet per second. The top of the ladder started to slid down.
How fast is the top of the ladder sliding down when it is 6 feet from the ground?
6. The product of two positive numbers is 48. What is the smallest possible value for the
sum of the first and thrice the second?
7. A Norman window has the shape of a rectangle surmounted by a semicircle. Thus, the
diameter of the semicircle is equal to the width of the the rectangle.) If the perimeter
of the window is 30 ft, find the dimensions of the window so that the greatest possible
amount of light is admitted.
8. A spherical ball bearing has a radius of 0.3 cm. However, there is an error of 0.01cm
in measuring the radius. Use differentials to find the error and the relative error of the
volume of the ball-bearing.
9. The revenue, R, is given by 100x − 3x2 − 0.1x3 for 0 ≤ x ≤ 120. Use differentials to find
the change in revenue when the sales go from x = 50 to x = 52.
10. Determine on which intervals the following functions are increasing or decreasing.
x−1
(a) y = x3 − x2
(b) y = −x3 + 3x2 − 2
(c) f (x) =
x−2
(d) y = xex
(e) y = x2 − x − 2
11. Determine on which intervals the following functions are concave upward or concave downward and find its point(s) of inflection. (a) y = 2x3 − 3x2 − 36x + 14
12. Determine all relative extremas of the following functions:
(a) f (x) = x4 − x3
4
(c) y = x +
x
(b) f (x) = 2x − 3x2/3
(d) y = x3 − 5x2 + 7x
1
(b) y =
x2
6
+3
13. Find the areas between the two curves y = f (x) and y = g(x) of the following:
(a)
f (x) = 9 − x2 ,
(b)
f (x) = x,
(c)
f (x) = 3 − x2 ,
g(x) = x2 − 9
√
g(x) = 3 x
g(x) = x2 + 5x
14. Evaluate the following
Z
Z
x2
5
(a)
dx
(b)
x4 ex dx
3
Z 1+x
Z
1
2x + 1
(d)
dx (e)
dx
−x
1+e
(x2 + x)2
Z
Z
3x2 + x − 2
4
(g)
x3 ex dx
(h)
dx
x2
(c)
(f)
(i)
Z
Z
Z
x2 ex dx (by parts)
y ln y dy (by parts)
3x2 + x − 2
dx
x−1
15. The marginal cost is 35 − 0.06x. Find the total cost of producing 200 units.
16. When a wholesaler sold a product at $40 per unit, sales were 300 units per week. After a
price increase of $5, the average number of units sold dropped to 275 per week. Find the
demand function and from it calculate what price per unit will yield a maximum total
revenue.
17. Evaluate the following improper integrals
Z ∞
Z ∞
Z ∞
1
x
3x
√
(a)
e
dx
(c)
dx
(b)
dx
2
x
1
5
0
x2 − 21
(d)
Z 1
0
1
dy
1−y
(e)
Z 9
0
√
1
dx
9−x
(f)
Z 4
0
1
√ dx
x
18. If I put $10000 in a bank that pays 6.5% p.a for interest, assuming interest are compounded
monthly, how much money would have have in the bank after 10 years? How much money
would I get if interest were compounded continuously?
19. If there are 10 fruit flies after 1 day and 150 fruit flies after 2 days on my rotten banana,
how many fruit flies would I see after 5 days (assuming an exponential growth model)?
20. What are the two main ingredients in a plate of Moo Goo Gai Pan?
21. Which asian country has a capital city with exactly three words? four words?
Z
dy
22. Which mathematician gave us the symbol
for the derivative of a function and f (x) dx
dx
for anti-derivatives?
23. Which city is Amos really from? Where did he earn his PhD?
24. What is ho dee hi minus hi dee ho over ho ho?
25. Who says “you can differentiate me all you want, I am . . .”?
26. Who invented Calculus and on which day was (s)he born?
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