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Remain Open - Calculus - Exam, Exams of Calculus

University of MumbaiCalculus

This is the Exam of Calculus which includes Start Biking, Square Inch, Some Points, Solution, Slope, Slope Fields etc. Key important points are: Remain Open, Material, Square Base, Maximum Possible Volume, Remain Open, Maximum, Justify, Equation, Implicit Differentiation, Minimum

Typology: Exams

2012/2013

Uploaded on 03/06/2013

ramesh
ramesh 🇮🇳

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NAME:
Math 105D - Exam 2 - November 10, 2006
Instructions: Show all of your work and circle your final answers. Calculators are allowed, but notes and
books are not.
1. (15 points) You have 54 ft2of material with which to build a box with a square base. One side (not
the top or the bottom) of the box is to remain open. What is the maximum possible volume of the
box? (Be sure to justify how you know this is a maximum.)
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NAME:

Math 105D - Exam 2 - November 10, 2006 Instructions: Show all of your work and circle your final answers. Calculators are allowed, but notes and books are not.

  1. (15 points) You have 54 ft^2 of material with which to build a box with a square base. One side (not the top or the bottom) of the box is to remain open. What is the maximum possible volume of the box? (Be sure to justify how you know this is a maximum.)
  1. (10 points) The equation x^3 y + y^3 = 7 + x defines a graph. Using implicit differentiation, find dydx.
  2. (10 points) Find the minimum and maximum values of the function f (x) = (^) 1 +x x 2 on the interval [− 3 , 2].
  1. (16 points)

(a) Calculate (^) xlim→∞^ e^2 x 3 x+ 2 x.

(b) Calculate lim x→π^ sin(3 x − xπ ).

  1. (10 points) For some function g(x), suppose we know that there is a root at x = a for some number a in the interval (1, 4). Also suppose that g(x) is increasing and concave down on the interval [1, 4]. (a) Draw the graph (labeled appropriately) of a function g(x) that satisfies the above properties.

(b) Using Newton’s method, if our initial guess x 0 is in [1, 4] and greater than a, will x 1 be less than a or greater than a, or can we not be sure? Explain. (Your graph above may be helpful.)

(c) What if our initial guess x 0 is in [1,4] and less than a? Explain.