Mathematics

Problem

The function

k(x)=\cos x

is differentiable for all real

x

. (a) Verify that the slope of the tangent line to the graph of

k

x=\tfrac{\pi}{4}

-\tfrac{\sqrt{2}}{2}

. (b) Describe the relationship between the slope of this tangent line and the slope of the corresponding normal line at

x=\tfrac{\pi}{4}

, and determine the exact value of the normal line’s slope. (c) Explain, without using numerical calculations, why the tangent line found in part (a) provides the best linear approximation to

\cos x

near

x=\tfrac{\pi}{4}

. Your explanation should reference relevant properties of differentiable functions.

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