Math 1100
Compiled Quizzes
14 December, 2013
Name:
Quiz 1
1. Evaluate each limit if it exists. If it does not exist, write DNE.
(a)
lim x2 − 7
x→4
(b)
(x − 4)(x + 1)
x→4 (x − 4)(x + 2)
lim
2. Let f (x) =
x + 6 if x ≥ 3
2x if x < 3
(a) Find f (3) if it exists.
(b) Find lim f (x) if it exists.
x→3
(c) Is f (x) continuous at x = 3? Explain your answer.
Quiz 2
3. Use the definition of a derivative (NOT derivative rules) to find the derivative of
f (x) = x2 + 1
Show all your work.
4. Find the equation of the line tangent to f (x) = x4 − x3 + 1 at the point (1,1). You may
use any methods you like.
Quiz 3
5. Find the derivative of each function.
(a)
√
f (x) = (2x + 1) x
(b)
g(x) =
(c)
h(x) =
2x
3x − 1
√
4x2 + 2
6. Let y =
1
x2
+ 6x2 .
(a) Find
dy
.
dx
(b) Find
d2 y
dx2
(c) Find
d3 y
dx3
7. Suppose the position of a particle is given by f (t) = 3t4 + 4t.
(a) Find the velocity of the particle at t = 1.
(b) Find the acceleration of the particle at t = 1.
Quiz 4
Let
1
1
f (x) = x3 − x2
3
2
8. Find f 0 (x).
9. Find f 00 (x)
10. Find all critical values of f (x).
11. Is f (x) increasing or decreasing at x = 2? Show clearly how you got your answer!
12. Is f (x) concave up or concave down at x = 2? Show clearly how you got your answer!
13. Find a local maximum point of f (x). Give both coordinates, not just the x-value.
Quiz 5
14. A restaurant can produce up to 60 hamburgers per day. The profit from the sale of
1 2
hamburgers is given by P (x) = − 30
x + 2x + 470. How many hamburgers should the
restaurant produce to maximize profit?
15. Differentiate each function:
(a)
f (x) = ln
6x + 3
x2 + 1
(b)
g(x) = x ln(x)
16. A farmer wants to fence a rectangular field of 800 ft2 . A river runs along one side of the
field, as shown below. What is the smallest length of fence required to fence the other
three sides?
Spacer!
Quiz 6
17. A spherical balloon is being inflated at the rate of 5 in3 /min. At what rate is the radius
increasing when the radius of the balloon is 5 inches? (You may wish to use one of these
formulas: V = 34 πr3 and A = 4πr2 .)
Quiz 7
18. Evaluate each integral.
(a)
Z
x5 + 7x dx
(b)
Z
1
dx
x2
(c)
Z
(x6 + 6x)3 (2x5 + 2) dx
(d)
Z
√
3
4x 2x2 + 1 dx
19. (Extra credit, 2 points) Find the general solution to
Z Z
2
x + 4 dx dx
Quiz 8
20. Evaluate each integral.
(a)
Z
4e4x dx
(b)
Z
2x
dx
+3
x2
21. If a certain firm has marginal cost given by M C = 6x + 60 and marginal revenue given
by M R = 180 − 2x where x is the number of units produced, how many units should
the company produce to maximize profit? (Remember to check that your solution gives
a maximum and not a minimum.)
22. Find the solution to
3y 2 dy
·
=1
2x dx
that passes through the point (2,1).
Your answer may be given in implicit form (that is, you do not need to solve for y.)
Quiz 9
23. Suppose the growth rate of a colony of bacteria is given by
dy
= 2y
dt
where y is the number of bacteria after t hours.
If there are 100 bacteria initially, how long will it take until there are 16,275,479 bacteria?
(You may assume that 16, 275, 479 = 100e12 .)
24. Consider the following graph representing the function y = x3 + x2 − 2x. (Grid lines are
one unit apart.)
3
2
1
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
(a) Write an indefinite integral representing the shaded area.
(b) Find the area of the shaded region.
Quiz 10
25. Find the area enclosed by the curves y = x2 − 1 and y = x + 1. (The two functions are
graphed below.) Make certain to show all your work.
3
2
1
-5
-4
-3
-2
-1
0
1
2
-1
-2
-3
26. Compute the integral if it exists. Show all work clearly.
Z ∞
−3
dx
x4
2
3
4
5
Quiz 11
27. Let f (x, y) = x2 ey + xy
(a) Compute f (2, 0).
(b) Compute fx .
(c) Compute fy .
(d) Compute fxy .
(e) What is the slope of the tangent line to f in the x-direction at the point (2,0)?
(f) What is the instantaneous rate of change of f with respect to y at (2,0)?