MATH 160 Final Exam, Fall, 2007 page 8 Name:
1. Use algebra and the limit theorems (NOT a calculator and NOT L’Hôpital’s Rule) to evaluate the following limits. If a limit does not exist or the limit is infinite, explain how you know.
Show details of your algebraic work.
(a) =
(b) =
(c) =
2. Use differentiation formulas to compute the indicated derivatives. Show details of how you used the differentiation formulas. Simplify as directed
(a) y = 5x3 + 6x7/5 – 3x-3/4 + 2p2 y′ = ______
y′′ = ______
(b) f(x) = (2x2 + 1) sinx f′(x) =
(c) s =
(d) y = sin2(6x2 + 1)
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) =
(b) =
(c) =
4. (a) Explain accurately in non-technical language what it means to say that a function y = f(x) has limit L as x approaches a.
(b) Evaluate the following limit numerically (including the possibility that the limit might not exist or be infinite). Organize the numerical work in a table. Explain clearly how the table you created and your understanding of what it means for a function to have (or not have) a limit leads to your conclusion.
5. Sketch the graph of a differentiable function that has all of the following properties:
x f(x) f′(x) f′′(x)
x < 0 f′(x) = 1
0 f(0) = 1 f′(0) = 1
0 < x < 2 f′(x) > 0 f′′(x) < 0
2 f(2) = 2 f′(2) = 0
2 < x < 3 f′(x) < 0 f′′(x) < 0
x = 3 f(3) = 0 f′(3) < 0 f′′(3) = 0
3 < x< 4 f′(x) < 0 f′′(x) > 0
x = 4 f(4) = -2 f′(4) = 0 f′′(4) > 0
4 < x f′(x) > 0 f′′(x) > 0
y
3 -
2 -
1 -
-3 -2 -1 0 1 2 3 4 x
-1-
-2-
-
6. Bob the Builder is working on the roof of a building and needs a board that is lying on the ground. The board is 20 feet long and lying perpendicular to the side of a building with one end against the building. Bob lifts the board up by a rope attached to the end of the board that is against the building. He carefully lifts the end of the board at the constant rate of 5 feet per second. How fast is the other end of the board moving toward the building when Bob has lifted the board 8 feet?
7. One definition of the derivative of a function f at
a number c is f'(c) =.
The figure shows the graph of a function y = f(x) and
a number c marked on the x-axis.
(a) Illustrate and label each of the quantities h, c + h,
f(c), and f(c+h) that appear in this definition on the graph. (You may assume h > 0.)
| x
c
(b) Illustrate (draw!) and explain how to interpret the number f'(c) in terms of the graph.
(c) Illustrate (draw!) and explain how to interpret the expression in terms of the graph.
(d) Explain in non-technical terms what the symbol means in the equation
f'(c) = that defines the derivative of a function f at a number c .
8. The position of a particle moving along a straight line path during the time interval 0 t 5 is given by
s(t) = t2 – t + 10.
(a) Find the average velocity of the particle over the time interval 0 t 5. Show clearly how you calculated its average velocity.
(b) Write the name and a complete statement of a theorem that tells us there must be at least one time when the instantaneous velocity of the particle is equal to the average velocity you found in (a).
(c) Find a time when the when the instantaneous velocity of the particle is equal to the average velocity you found in (a).
(d) Which direction (toward or away from the origin) is the particle moving at time t = .25 sec? at time t = 1 sec? Show clearly how you know.
(e) Find the total distance (not the displacement) the particle traveled during the time interval 0 t 5. Show clearly how you found this distance.
(f) Show that during the time interval 0 t 5 the acceleration of the particle is constant.
9. (a) Write a definite integral that gives the area of the region enclosed by the graph of y = 1 – cos(px) and the
x-axis between x = 0 and x = 2.
(b) Find the exact area of the region in (a).
Aaron and LeAnn want to calculate a Riemann sum using 4 subintervals of equal length to calculate an approximate value for the area of the region described above. They are trying to decide what evaluation points to use. Aaron wants to use right endpoints. LeAnn is arguing for using midpoints because, she says, midpoints will give a better approximation for the area.
(c) Explain why one would expect a Riemann sum using midpoints as evaluation points to give a better approximation for the definite integral than a Riemann sum using endpoints.
(d) Write out explicitly and calculate the numerical value of the Riemann sum Aaron proposes (4 equal subintervals, right endpoints as evaluation points).
(e) Use the graph to explain why Aaron’s Riemann Sum gives such a good approximation for the area.
10. Consider the region enclosed by the graphs of the straight line y = (5/7) + (1/7)x and the semicircle
y = .
(a) Show that the graphs of the straight line y = (5/7) + (1/7)x and the semicircle
y = intersect at the points (-4/5, 3/5) and (3/5, 4/5). Show details of the verification.
(b) Write an integral that gives the area of this region. (Do not evaluate the integral.)
(c) Write an integral that gives the volume of the solid formed by revolving this region around the x-axis.
(Do not evaluate the integral.)
(d) Write an integral that gives the volume of the solid formed by revolving this region around the vertical line x = – 1. (Do not evaluate the integral.)
11. A thin rod of variable density lies along the interval 0 x 2 on the x-axis. The linear density of the rod at a point x-units from the origin is d(x) = .
(a) Find the total mass of the rod.
(b) Find the moment of the rod around the origin.
12. (a) Write an integral that gives the arc length of a semicircle or radius 1.
(b) Use the result from (a) to conclude that = p.