Homework 6: Rules for differentiation

Differentiation exercises

Exercise 6.1 : Find [Maple Math] of


(a) [Maple Math] ,

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(b) [Maple Math] ,

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(c) [Maple Math] (expand out the cube first),

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(d) [Maple Math] (use the chain rule),

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(e) [Maple Math] .

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(f) [Maple Math] ,

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(g) [Maple Math]

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(h) [Maple Math] ,

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(i) [Maple Math] (use the chain rule),

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(j) [Maple Math] (factor out the [Maple Math] and use the product or quotient rule),

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(k) [Maple Math] ,

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(l) [Maple Math] ,

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(m) [Maple Math] ,

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(n) [Maple Math] .
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Exercise 6.2 : Find the first derivative [Maple Math] and also the second derivative [Maple Math] of the following functions of [Maple Math] (here a, b, and c are constants):

(a) - [Maple Math]

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(b) [Maple Math]

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(c) [Maple Math]

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(d) [Maple Math] .
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Story Problems using the derivative

Exercise 6.3 : Suppose that the bicycle in Example 8 has distance function [Maple Math] . Find [Maple Math] and [Maple Math] . Graph x(t) and v(t), and explain in words what the bicycle is doing.


[Maple Plot]

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Exercise 6.4: If [Maple Math] , then [Maple Math] . Use the tangent line approximation to find [Maple Math] . Express your answer in terms of n and [Maple Math] .
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Exercise 6.5 : You want a ladder to reach a window which is about 24 feet above the ground. You place the bottom of the ladder exactly 7 feet away from the inside of the building. If the window were exactly 24 feet high, you compute that you would need 25 feet of ladder to reach it. Suppose that the error in your value of 24 for the height of the window is +/- h. Find the error in the figure of 25 feet for the length of ladder.

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Exercise 6.6 : In the situation of Problem 6 of the last homework, suppose that you want to fit the triangle in the smallest possible sphere (rather than circle). In that case the triangle's hypotenuse becomes a diameter of the sphere. Find the volume of that smallest sphere, and include an error estimate obtained using the tangent line approximation.
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Exercise 6.7: Suppose that an object falling from a height of 80 meters (see Problem 4 of Chapter 3) is viewed from a height of 2 meters by a person standing 10 meters from the side of the building where the object is falling. (a) Find a formula in terms of t for the distance from the person's eye to the object. (b) Find a formula in terms of [Maple Math] for the rate at which that distance is changing.
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