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Calculus I - Answer the Questions, Exam 3 | MATH 161, Exams of Calculus

Material Type: Exam; Professor: Stout; Class: Calculus I; Subject: Mathematics; University: Illinois Wesleyan University; Term: Fall 2003;

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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bg1
Math 161, Fall 2003
Exam 3
Name:
Answer all questions.
1. Prove from the definitions that if f0(a)>0 then fdoes not have a local maxi-
mum at a. (15 Points)
2. Prove that if f0(x) = g0(x) on (a, b) then f(x)g(x) is constant on (a, b).
(You may quote, rather than proving, the relevant corollary to the Mean Value
Theorem from class.) (10 Points)
3. Find the maximum and minimum of
g(x) = x2 |x|
on [1,2] (15 Points)
4. You are given a circular piece of plastic with radius 10 cm. How do you make
a conical drinking cup from it with maximum volume? (15 Points)
5. Find the function to be minimized and the interval on which to minimize it for
the following situation: What are the proportions of the pyramid with a square
base with a given volume which lead to least total surface area? (10 Points)
6. Where is f(x) = (x2)(x4)
x21increasing? (10 Points)
7. Where is h(x) = x3
x2+ 1 concave up? (10 Points)
Turn Page Over
1
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Math 161, Fall 2003

Exam 3

Name: Answer all questions.

  1. Prove from the definitions that if f ′(a) > 0 then f does not have a local maxi- mum at a. (15 Points)
  2. Prove that if f ′(x) = g′(x) on (a, b) then f (x) − g(x) is constant on (a, b). (You may quote, rather than proving, the relevant corollary to the Mean Value Theorem from class.) (10 Points)
  3. Find the maximum and minimum of

g(x) = x^2 − |x|

on [− 1 , 2] (15 Points)

  1. You are given a circular piece of plastic with radius 10 cm. How do you make a conical drinking cup from it with maximum volume? (15 Points)
  2. Find the function to be minimized and the interval on which to minimize it for the following situation: What are the proportions of the pyramid with a square base with a given volume which lead to least total surface area? (10 Points)
  3. Where is f (x) =

(x − 2)(x − 4) x^2 − 1

increasing? (10 Points)

  1. Where is h(x) =

x^3 x^2 + 1

concave up? (10 Points)

Turn Page Over

  1. Sketch a graph using the following information: (15 Points)

Asymptotes: y = −1 (to left), y = 1 (to right) x = 1 Values and signs: f (−2) = − 3 , f (−3) = − 2 f(x) - - - + + f’(x) - - + + - f”(x) - + + + + x -3 -2 0 1

Draw your graph here: