M160 Exam 3 Part A Calculator Required Name: ______
1. (7 points) If possible, sketch the graph of an increasing function such that f′′(x) changes from positive to negative at x = – 1 and from negative to positive at x = 2. If there is no such function, explain how you know.
y
-3 -2 -1 0 1 2 3 4 x
2. (8 points) The critical points of f(x) = x2 are x = , x = 0, and x = ( ≈ 2.45).
(a) On what interval (or intervals) is this function increasing?
Show in detail how one can use calculus to tell without looking at the graph that the function is increasing on this (these) interval(s). (Of course you can look at the graph to check your answer.)
(b) Does this function have a local maximum, a local minimum, or neither at the right endpoint of its domain? Show in detail how one can use calculus to tell without looking at the graph. (Again, use the graph to check your answer.)
3. (10 points) The graph of the function f(x) =
over the interval [ –1, 1 ] is shown in the figure at right.
All parts of this problem refer to this function and this graph.
1
-1 0 1
(a) What is the exact value of ? Explain clearly how you know.
(b) Partition the interval [ –1, 1 ] on the x-axis into 6 subintervals of equal length. On the figure above, mark these partition points and label them x0, x1, x2, etc. List the values of the partition points.
(c) Write an expression without using sigma notation (in expanded form) for the specific Riemann sum for the function f(x) = over the interval [ –1, 1 ] formed by using the partition from (b) and midpoints as evaluation points. Use the symbol f or the expression for the function (not the numerical values of the function) and the numerical values of the evaluation points in your Riemann sum.
(d) Calculate the numerical value of the Riemann sum you wrote in (c). (Use your calculator. Write your answer to the full accuracy of the calculator. Do not round your answer.)
(e) Illustrate in the graph above (draw!) and explain (write about what you drew!) how to interpret the Riemann sum you wrote in (c) graphically.
(f) How would you choose a partitioning { x0, x1, x2,…, xn } of [ –1, 1 ] so that no matter where the evaluation points { c1, c2, …, cn } are located the Riemann sum computed from this partitioning of [ –1, 1 ] will be very close to the exact numerical value of ?
4. (10 points) A box with square top and bottom and volume 20 cubic feet is to be constructed from two different materials. Exotic hardwood costing $5 per square foot will be used for the top and bottom. Polished metal costing $2 per square foot will be used for the sides. Find the dimensions of the box for the total cost of these materials to be as small as possible.
(Your solution will be graded using the criteria handed out in class.)
5. (10 points) Clarence is using Newton’s method to find by finding the zero of the function y = x3 + 2.
(a) Sketch an accurate graph of the function y = x3 + 2
on the coordinate system at right.
2 –
1 –
-2 -1 0 1 2
(b) Clarence takes xo = 1 as his initial approximation. Find an equation for the line Clarence should use to find the next Newton approximation, x1 (and show clearly how you used calculus to find it).
Sketch this line on the graph you drew for part (a).
(c) Use the equation you wrote in (b) to find the next Newton approximation, x1. Show your work!
(d) Draw the line Clarence should use to calculate x2 from x1.
What appears to be happening? What advice would you give Clarence?
M160 Exam 3 Part B Calculator Not Allowed Name: ______
6. (25 points) Evaluate the following indefinite and definite integrals. Show details of your calculations.
(a) =
(b) =
(c) =
(d) =
(e) =
7. (10 points) At time t = 0 a car traveling with velocity 96 ft/sec begins to slow down with constant deceleration
a = – 12 ft/sec2.
(a) Find the velocity v(t) of the car at time t .
(b) Find the distance the car traveled before it came to a stop.
8. (10 points) (a) Show that = + C.
(b) Show that = + C.
(c) Explain how an indefinite integral can have two such different values.
9. (10 points) The domain of the function F(x) = is – 1 x 1.
Find the following. (Note that the graph of appears in problem 1.)
(a) F(0) = ______
(b) F(1) = ______
(c) F(–1) = ______
(d) F¢(x) = ______
(e) F¢(3/5) = ______