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Solution - 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: Solution, Limit Definition, Derivative, Credit, Equation, Line Tangent, Graph, Differential Equation, Initial Value Problem, Filled

Typology: Exams

2012/2013

Uploaded on 03/06/2013

ramesh
ramesh 🇮🇳

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Math 105C Test 1 (75 points)
Name:
•Check that you have 8 questions on three pages.
•Show all your work to receive full credit for a problem.
1. (10 points) Let f(x)=5x−x2.
(a) Use the limit definition of the derivative to find f0(1).
(b) Find the equation of the line tangent to the graph of fat x=1.
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Math 105C Test 1 (75 points)

Name:

  • Check that you have 8 questions on three pages.
  • Show all your work to receive full credit for a problem.
  1. (10 points) Let f(x) = 5x − x^2.

(a) Use the limit definition of the derivative to find f′(1).

(b) Find the equation of the line tangent to the graph of f at x = 1.

  1. (9 points) Show that y = x −

x

is a solution of the differential equation x^2 y′′^ + xy′^ = y.

  1. (9 points) Solve the IVP (initial value problem):

x^3

− f′(x) = 3x + 1; f(1) = 4.

  1. (10 points) A tank is filled with 600 gallons of water. Water starts flowing out of the tank after a drain at the bottom of the tank is opened and the volume of water, V , in the tank t minutes after the drain is opened is given by the function V (t) = 600 − 20 t +

t^2. Include units in your answers.

(a) Find the average rate of change of the volume of water from time t = 15 minutes to time t = 45 minutes.

(b) Find the instantaneous rate of change of the volume of water at time t = 15 minutes and at time t = 45 minutes. At what instant is the volume decreasing more rapidly - at t = 15 or at t = 45? Justify your answer.