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Math 181 Test 2: Derivatives, Rates of Change, and Optimization, Exams of Mathematics

Iowa State University (ISU)Mathematics

Math 181 test 2 questions covering topics on calculus, including finding derivatives, rates of change, local maxima and minima, and optimization. Questions involve calculating derivatives of various functions, determining rates of change for biomass and fungus growth, and setting up equations for optimizing the cost of fencing a rectangular field.

Typology: Exams

Pre 2010

Uploaded on 09/02/2009

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Name:
Math 181
Test 2
1. Calculate the derivatives of the following functions.
a) f(x) = x4
−4x3+3
√x+1
x2+π2
b) f(x) = (x2+ 3x+ 1)(3x4
−5x3+ 1)
(You do not need to simplify the answer.)
c) f(x) = 2x−1
x3+ 1
d) f(x) = √4x2
−2x+ 5
e) Find f′′(x) of f(x) = 3x4+ 2x3+x−5.
1
pf3

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Download Math 181 Test 2: Derivatives, Rates of Change, and Optimization and more Exams Mathematics in PDF only on Docsity!

Name: Math 181 Test 2

  1. Calculate the derivatives of the following functions. a) f (x) = x^4 − 4 x^3 + 3

x +

x^2

  • π^2

b) f (x) = (x^2 + 3x + 1)(3x^4 − 5 x^3 + 1) (You do not need to simplify the answer.)

c) f (x) = 2 x − 1 x^3 + 1

d) f (x) =

4 x^2 − 2 x + 5

e) Find f ′′(x) of f (x) = 3x^4 + 2x^3 + x − 5.

  1. The biomass of a bacterial population after t hours is given by P (t) = − 2 t^2 + 100t + 50 mg. a) How much did the mass grow between hours t = 1 and t = 2? b) Calculate the average rate of change between hours t = 1 and t = 2. c) What is the instantaneous rate of change at hour t = 1?
  2. A fungus is growing in the shape of a sphere. When t = 3 days, its radius is 5 cm and its volume is increasing at the rate of 2 cm^3 /day. a) How fast is its radius changing? (Give your answer to 6 decimal places.) b) Is the radius increasing or decreasing?