The function is differentiable for all real .

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

The function k(x)=cos⁡xk(x)=\cos x is differentiable for all real xx. (a) Verify that the slope of the tangent line to the graph of kk at x=π4x=\tfrac{\pi}{4} is −22-\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=π4x=\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\cos x near x=π4x=\tfrac{\pi}{4}. Your explanation should reference relevant properties of differentiable functions.

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