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Circular Shape - Multivariable - Exam, Exams of Calculus

Amity University - BiharCalculus

main points of this exam paper are: Circular Shape, Vegetable Oil, Cubic Inches, Minute, Shallow Puddle, Maintaining, Expanding, Constant Height, Mean Value Theorem, Guaranteed

Typology: Exams

2012/2013

Uploaded on 03/21/2013

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Math 105 A and B INITIALS
Final Exam, page 1. April 6, 2006
1. Vegetable oil spills from a hole in a cup at 0.5 cubic inches per minute and forms a shallow puddle
of constant height 1/16 inch while maintaining a circular shape. When the circle has an eight-inch
radius, what is the rate at which it is expanding?
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  1. Vegetable oil spills from a hole in a cup at 0.5 cubic inches per minute and forms a shallow puddleof constant height 1/16 inch while maintaining a circular shape. When the circle has an eight-inch radius, what is the rate at which it is expanding?

2a. State the Mean Value Theorem completely.

2b. Find the c guaranteed to exist by the MVT for f(x) = √x on [1, 4]. Show all your work.

4a. Use summation, or sigma, notation to express the Riemann sum Ln for

1

√x dx.

4b. Explicitly show the numbers in the sum L 6 and compute their sum.

4c. Find the exact value of

1

√x dx using the FTC.

4d. Find L 20 , M 20 , R 20 and T 20 for

1

√x dx. Express each answer to six places after the decimal point.

4e. What is the average rate of change of f(x) = √x on the interval [1, 4]?

4f. What is the average value of f(x) = √x on the interval [1, 4]?

  1. Consider the initial value problem (IVP)

dy y^ dx(1) = 4^ = 0.^5 y. 5a. For what values of A and B does y = AeBx^ satisfy the differential equation dy dx = 0. 5 y?

5b. For what values of A and B does y = AeBx^ satisfy the IVP as given?

  1. Find dy/dx for each of the following. a. y = arctan √sin(x^3 )e^2 x

b. y = ln(x^2 + 2x^ + 2^2 + x−^4 + (3/x) + e^3 )

c. 3xy^ + x^3 + y^4 = tan(3x + 5)

  1. Use geometry or a version of the FTC or perhaps both to find each of the following: a.

1 20 w

(^4) + 10w dw

b.

0 4 +^

√ 9 − x (^2) dx.

c. (^) dxd

( (^) d dx

(∫ (^) x 1 ln(t

(^2) + cos t) dt^ ))