Chapter 4 Applications of Derivatives

MATH 160 Exam 3 Fall, 2007 Part A: Calculator Allowed Name:

1. Use calculus to solve the following problem. A car is traveling 84 ft/sec. At time t = 0 the driver applies the brake and decelerates at a constant rate of 14 ft/sec2 until coming to a stop. (a) How long does it take for the car come to a stop?

(b) How far will the car have traveled before coming to a stop? MATH 160 Exam 3 Fall, 2007 Part A: Calculator Allowed Name:

2. LeAnn is explaining to Aaron how Newton’s method works by showing graphically how to find the successive Newton approximations to the smallest positive solution to the equation sin(3x) – cos(x) = 0 starting with

x0 = 0.5 as the first approximation to the solution. (Be sure to use RADIAN mode!) (a) Sketch a graph that LeAnn could use to explain to 1– Aaron how to solve this equation.

(b) In your figure, mark and label the starting point x0 = 0.5.

______0 0.5 1 1.5 2

(c) Sketch the line used to find the next Newton approximation x1. Mark the Newton approximation x1 on your drawing.

(d) Use Newton method to find the next two approximations x1 and x2 to the solution.

x1 = ______

x2 = ______

Give x1 and x2 to at least 8 decimal places. Details of the calculations are not required, but if you give details they must be correct. MATH 160 Exam 3 Fall, 2007 Part A: Calculator Allowed Name:

x 3. Aaron and LeAnn are discussing the function y = F(x) defined for -3 ≤ x ≤ 3 by F(x) = 9  t 2 d t . 0 (a) Aaron used his calculator and found that, rounded to 6 decimal places, F(3) = 7.068605. LeAnn looked at Aaron’s calculator and said excitedly “Hey, you didn’t need to do that! It’s easy to get the exact value of F(3) from the graph you have on your calculator screen!”  What graph did LeAnn see on the screen of Aaron’s calculator? How could she get the exact value of F(3) from what she saw? What is the exact value of F(3) ?

(b) For what values of x is F(x) negative? Explain why F(x) negative for these values of x. (“Because I graphed F(x) on my calculator and can see where F(x) is negative” does not explain why!)

(c) Are there any values of x for which F(x) = 0 ? If not, explain/show how you know there are not. If so, find all such values of x and explain why the function is zero at the x value(s) you found.

(d) The derivative of the function y = F(x) is F(x) = ______

(e) Take g(x) to be the function g(x) = tan x . Find the derivative of the composite function Fog(x) = F(g(x)). MATH 160 Exam 3 Fall, 2007 Part A: Calculator Allowed Name:

4. (a) Show without relying on the graph that the points x1 = 0 and x2 = 1 are critical points of the function f(x) = 2x1 3  2x2 3  x2 . (These are the only critical points.)



(b) Use EITHER the first derivative test OR the second derivative test to determine without relying on the graph whether the function has a local maximum, a local minimum, or neither at the critical point x1 = 0. Show the details of your work and explain clearly how your work leads to your conclusion.

(c) Use EITHER the first derivative test OR the second derivative test to determine without relying on the graph whether the function has a local maximum, a local minimum, or neither at the critical point x2 = 1. Show the details of your work and explain clearly how your work leads to your conclusion.

(d) Graph the function on your calculator. Explain what you see in the graph that tells you x1 = 0 is a critical point of this function.

(e) Explain what you see in the graph that tells you x2 = 1 is a critical point of this function.

(f) Does it appear from the graph that this function has an absolute maximum? If so, what do you see in the graph that indicates to you that the function has absolute maximum? If not, what do you see in the graph that indicates to you that the function doesn’t have an absolute maximum? MATH 160 Exam 3 Fall, 2007 Part A: Calculator Allowed Name:

5. Betty plans to build an open topped box 3 inches deep by cutting squares of length h = 3 from the four corners of a rectangular piece of cardboard and folding up the sides. She wants the outside of the box to have area 108 in2, so she plans to use a rectangular piece of cardboard with area 144 in2. She also wants the volume of the box be as large as possible. (Outside of the box consists of four sides and the bottom.) (a) What should be the dimensions of the piece of cardboard Betty uses to build this box? (b) If Betty builds the box from a piece of cardboard of the size you recommend in answer to (a), what will be the volume of the box Betty builds? MATH 160 Exam 3 Fall, 2007 Part B: Calculator NOT Allowed Name:

6. The figures show graphs of the first derivative f ′ ( x) and second derivative f ′ ′ ( x) of a function y = f ( x).

Graph of y = f ′(x) Graph of y = f ′′(x)

(a) On what interval(s) is the graph of y = f ( x) decreasing? ______

(b) On what interval(s) is the graph of y = f ( x) concave up? ______

(c) The graph of the function y = f ( x) described above includes the point ( 0, –1). Sketch the graph of this function. MATH 160 Exam 3 Fall, 2007 Part B: Calculator NOT Allowed Name: 7. Evaluate the following integrals. Give specific numerical values for definite integrals.

3 2 3 5 3 5 (a)  8x  x 16x  4x dx 

 (b) 4sin  3cos  d = (Evaluate any trigonometric functions that may appear in your answer.) 0

2 (c)  sec x tan x  sec x dx 

3 (d)  ( t  t )(3t  4) dt

2 (e) 2  | x | dx  1 MATH 160 Exam 3 Fall, 2007 Part B: Calculator NOT Allowed Name: b 8. The Riemann definite integral f ( x ) d x is defined to be the limit of Riemann sums:  a b f ( x ) d x = lim { f(c1) x1 + f(c2) x2 + f(c3) x3 + … + f(cn-1) xn-1 + f(cn) n }  a where x1 = (x1 – x0 ), x2 = (x2 – x1 ), x3 = (x3 – x2 ), … , xn-1 = (xn-1 – xn-2 ), and xn = (xn – xn-1 ). A function y = f(x) defined on the interval [a, b] = [-1, 2] is shown in the figure. 

(a) Explain in words AND illustrate in the figure how to interpret x0, x1, x2, x3, … , xn-1, xn

(b) Explain in words AND illustrate in the figure how to interpret c1, c2, c3, … , cn-1, cn.

(c) Explain in words AND illustrate in the figure how to interpret the product f(c2) x2 = f(c2) (x2 – x1 ).

(d) Explain in words AND illustrate in the figure how to interpret the sum

f(c1) (x1 – x0 ) + f(c2) (x2 – x1 ) + f(c3) (x3 – x2 ) + … + f(cn-1) (xn-1 – xn-2 ) + f(cn) (xn – xn-1 )

(e) You know from work you did earlier in this course that a limit means some number can be approximated as accurately as we want by controlling something else that’s used to calculate the approximations. Use the figure to illustrate (draw!) and explain how to interpret the number that can be approximated arbitrarily accurately in the definition of the Riemann integral. What must one do to be sure that computed Riemann sums give an accurate approximation for this quantity?