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Practice Test 2 - University Calculus II - Spring 2009 | MTH 1740, Exams of Calculus

Material Type: Exam; Professor: Rosenthal; Class: UNIVERSITY CALCULUS II; Subject: MATHEMATICS; University: St. John's University-New York; Term: Spring 2009;

Typology: Exams

Pre 2010

Uploaded on 08/17/2009

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MTH 1740: PRACTICE TEST 2
University Calculus II
Spring 2009
Dr. Rosenthal
DISCLAIMER: This practice test is a guide to organizing your studies and an indication of the
difficulty of the test problems, as well as the amount of time required to finish the test. You are
absolutely responsible for all of the class notes and homework for the test. You may not conclude
that a certain topic is not on the coming test just because it is not on the practice test. You may
not conclude that a certain topic is on the coming test just because it is on the practice test.
1. (a) Simplify the expression: tanh(ln x).
(b) Differentiate y= cosh(x31).
(c) Prove that the function y=f(x) = Ccosh 5x+Dsinh 5x, where Cand Dare
constants, is a solution of the differential equation y00 25y= 0.
(d) Evaluate Z1
cosh xsinh xdx.
2. Sketch the polar equation r= 1 + 2 cos 2θ.
3. Find the distance between the points with polar coordinates (r, θ)=(2,7π
4) and
(r, θ) = (2,π
6).
4. Write down an integral that will calculate the area of one of the petals of the 8-petal
rose, r= sin 4θ.
1
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MTH 1740: PRACTICE TEST 2

University Calculus II Spring 2009 Dr. Rosenthal

DISCLAIMER: This practice test is a guide to organizing your studies and an indication of the difficulty of the test problems, as well as the amount of time required to finish the test. You are absolutely responsible for all of the class notes and homework for the test. You may not conclude that a certain topic is not on the coming test just because it is not on the practice test. You may not conclude that a certain topic is on the coming test just because it is on the practice test.

  1. (a) Simplify the expression: tanh(ln x).

(b) Differentiate y = cosh(x^3 − 1).

(c) Prove that the function y = f (x) = C cosh 5x + D sinh 5x, where C and D are constants, is a solution of the differential equation y′′^ − 25 y = 0.

(d) Evaluate

cosh x − sinh x dx.

  1. Sketch the polar equation r = 1 + 2 cos 2θ.
  2. Find the distance between the points with polar coordinates (r, θ) = (−√ 2 , 74 π ) and (r, θ) = (2, π 6 ).
  3. Write down an integral that will calculate the area of one of the petals of the 8-petal rose, r = sin 4θ. 1

2 MTH 1740: PRACTICE TEST 2

  1. A hawk flying at an altitude of 180m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation y = 180 − x

2 45 until it hits the ground, where y is its height above the ground and x is the horizontal distance traveled in meters. Set up, but do not evaluate, an integral that will calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground.

  1. Set up, but do not evaluate, an integral for the volume of the solid generated by rotating the region bounded by the curves y = 0, y = tan x, x = 0 and x = π/ 4 about the vertical line x = −1.
  2. A certain small country has $10 billion in paper currency in circulation, and each day $50 million comes into the country’s banks. The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks. Let c(t) denote the amount of new currency in circulation at time t, with c(0) = 0. (a) Formulate a mathematical model, i.e., a differential equation, that represents the “flow” of the new currency into circulation. (Hint: This is a mixing problem, where the total rate of change is the rate coming in minus the rate going out.) (b) Solve the initial-value problem found in part (a).

Bonus Using the Principle of Mathematical Induction, prove that 2n+3^ ≤ (n + 3)! for all integers n ≥ 1.