   Chapter 3.10, Problem 42E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# On page 431 of Physics: Calculus, 2d ed., by Eugene Hecht (Pacific Grove, CA: Brooks/Cole, 2000), in the course of deriving the formula T = 2 π L / g for the period of a pendulum of length L, the author obtains the equation aT = –g sin θ for the tangential acceleration of the bob of the pendulum. He then says, “for small angles, the value of θ in radians is very nearly the value of sin θ; they differ by less than 2% out to about 20°.”(a) Verify the linear approximation at 0 for the sine function:sin x ≈ x(b) Use a graphing device to determine the values of x for which sin x and x differ by less than 2%. Then verify Hecht’s statement by converting from radians to degrees.

(a)

To determine

To verify: The linear approximation to the sine function at x=0

Explanation

Given:

The value f(x)=sinx.

Result used:

The linear approximation of the function at x=a is,f(x)f(a)+f(a)(xa).

Calculation:

The linearization of the function f(x) at x=0 is computed as follows,

Substitute the value a=0 in f(x)f(a)+f(a)(xa),

f(x)f(0)+f(0)(x)

The derivative of the function f(x)=sinx is, f(x)=cosx

(b)

To determine

The values of x for which sinx and x differ by less than 2% and verify that the Hecht’s statement by converting from radians to degrees.

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