Name ___________________
Student ID _______________
Date ____________________________________
TAKE HOME Test #3
(100 points Total)
Math 5A
Professor O. LePoint
Instructions:
Answer the following questions at home. You can use your notes and your book. You are expected to work independently on this test otherwise a ZERO score will be given. Write all steps CLEARLY and show all work. Full steps and solutions will gain full credit.
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QUESTION 1 ( 5 points) .
Let [pic]
a) Find the all critical points. b) Identify all inflection points.
QUESTION 2 ( 5 points).
Evaluate: [pic]
QUESTION 3 ( 10 points):
Suppose you are a design engineer who needs to create a metal box for the aft of a miniature engine. Suppose you wanted to make an open-topped box out of sheet medal that is 25" long by 20" wide. You machine a square out of each corner, all the same size, then fold up and weld the metal flaps together to form the rectangular box, as illustrated below. How big would you machine -out squares in order to maximize the volume of the box. State your equations, your optimization step, and your conclusion in clear terms.
[pic]
QUESTION 4 ( 10 points):
Suppose that you wanted to find the [pic] by using Newtons’ Method.
a) Define a function [pic]for Newton’s Method. (Of course, this is easy if you have a calculator, but here is a hint: Think of [pic]as a solution to some function f(x).)
b) Show the method to find [pic]such what when Newton’s method is applied, [pic].
Question 5) (5 points)
c) [pic]
Use the figure above, and draw a point (c,0), and its tangent line to mathematically explain the Mean Value Theorem and average rate of change between x and x+h.
QUESTION 6: (10 points)
Differentiate the following. Remember to simply your result by factoring out any greatest common factor.
a) [pic]. Find [pic].
b) [pic] Find [pic]
QUESTION 7 ( 10 points )
Find the x = c that satisfies the Mean Value Theorem for the function f(x) = x3 with endpoints x = 0 and x = 2.
QUESTION 8 ( 10 points):
Find two nonnegative numbers whose sum is 9, and so that the product of one number and the square of the other number is a maximum.
