M160 Final Exam Study Guide - Open Computing Facility
1. Use algebra and the limit theorems (NOT a calculator and NOT l’Hopital’s Rule) to evaluate the following limits. Show details of your work. If a limit does not exist or the limit is infinite, explain how you know.
(a) [pic][pic] =
(b) [pic][pic] =
(c) [pic][pic] =
2. Use differentiation formulas to compute the indicated derivatives. Show details of how you used the differentiation formulas.
(a) f(x) = [pic] f′(x) =
(b) s = [pic] [pic]
and (not simplified) [pic]
(c) g([pic]) = [pic] g′([pic]) =
3. Evaluate the following integrals by using integration techniques and/or antidifferentiation formulas (not a calculator). Show details of how you used integration techniques and/or antidifferentiation formulas.
(a) [pic] =
(b) [pic] =
(c) [pic] =
4. Sketch the graph of a differentiable function that has all the following properties:
x f(x) f’(x) f’’(x)
x < 0 f’(x) > 0 f’’(x) < 0
0 f(0) = 1 f’(0) = 0
0 < x < 2 f’(x) < 0 f’’(x) < 0
2 f(2) = -1 f’(x) < 0 f’’(2) = 0
2 < x < 3 f’(x) < 0 f’’(x) > 0
x = 3 f(3) = -2 f’(3) = 0 f’’(3) = 0
3 < x f’(x) < 0 f’’(x) < 0
y
-3 -2 -1 0 1 2 3 4 x
5. (a) We say that a function y = f(x) is continuous at x = a to mean that _____________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
(b) If possible, give a graphical example
of a function that is not continuous at
the point a = 1 but does have a limit
as x approaches a = 1.
If it is not possible to give such an
example, explain why.
-1 1 2
(c) If possible, give a graphical example
of a function that is not continuous at
the point a = 1 but is defined at the
point a = 1.
If it is not possible to give such an
example, explain why.
-1 1 2
(d) If possible, give a graphical example
of a function that is not continuous at the
point a = 1 but has both a left-hand and
a right-hand limit as x approaches a = 1.
If it is not possible to give such an
example, explain why.
-1 1 2
6. (a) Use calculus (not a calculator) to find all the points where the function y = f(x) defined on the interval
0 < x < 2 by f(x) = [pic] could have an extremum. List these points and show clearly how you found them.
(b) Choose one of the points you listed in (a) and use the first derivative test to determine whether the function has a local maximum, a local minimum, or neither at the point you chose. Show clearly how you applied the first derivative test and how it leads to your conclusion.
(c) Choose another of the points you listed in (a) and use the second derivative test to determine whether the function has a local maximum, a local minimum, or neither at the point you chose. Show clearly how you applied the second derivative test and how it leads to your conclusion.
7. Air is being pumped into a spherical balloon at the constant rate of 200 cm3/sec. How fast is the radius of the balloon changing at the instant when the radius is 5 cm?
8. (a) Sketch an accurate graph of the function
f(x) = x |x|.
(b) From examining the graph, does it appear
that the function f(x) = x |x| is differentiable
at the point c = 0? Describe what you see in
the graph that tells you.
-2 -1 1 2
(c) State the mathematical definition of derivative of a function y = f(x) at a point x = c.
(d) Use the definition from (c) to show that your conclusion in (b) is correct.
9. (a) Find the exact area of the region between the curve y = 1 + sin((x) and the x-axis on the interval
0 ( x ( 2 using a definite integral. Show clearly how you (not our calculator) evaluated the integral.
(b) Find an approximation for the area of the region in (a) by using a Riemann sum with four (4) rectangles
and left endpoints as evaluation points. Use the figure below to illustrate and explain how to interpret the terms in the Riemann sum graphically.
[pic]
(d) How do you explain the fact that the Riemann sum calculated in (b) gives such a good approximation for the area of the region when the number of approximating rectangles is so small?
10. (a) Find the total mass of a thin rod of linear density ((x) = x + [pic] gm/cm lying along the interval
1 < x < 3 of the x-axis. Show clearly how you evaluated the integral without using your calculator.
(b) Find the moment around x = 0 of a thin rod of linear density ((x) = x + [pic] gm/cm lying along the interval 1 < x < 3 of the x-axis. Show clearly how you evaluated the integral without using your calculator.
11. (a) Write an integral that represents the volume
of the solid obtained by rotating the region
enclosed by the curve y = x – x9, the vertical
lines x = 0 and x = 1, and the x-axis around
the x-axis. Sketch the region and illustrate a
typical volume element used to calculate the
volume. _____ __| |_____
DO NOT EVALUATE THE INTEGRAL. -1 0 1
(e) Write an integral that represents the volume
of the solid obtained by rotating the region
enclosed by the curve y = x – x9, the vertical
lines x = 0 and x = 1, and the x-axis around
the y-axis. Sketch the region and illustrate a
typical volume element used to calculate the
volume. _____ __| |_____
DO NOT EVALUATE THE INTEGRAL. -1 0 1
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