Let , defined for all real . The expression R(h)= h ( 4 +h )-h ( 4 ) h , h 0 depends…

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

Problem

Let h(x)=cos⁡x exh(x)=\cos x\,e^{x}, defined for all real xx. The expression R(h)=h(π4+h)−h(π4)h,h≠0R(h)=\frac{h\bigl(\tfrac{\pi}{4}+h\bigr)-h\bigl(\tfrac{\pi}{4}\bigr)}{h},\qquad h\neq0 depends on the small increment hh. (a) Compute lim⁡h→0R(h)\displaystyle \lim_{h\to0} R(h) without simplifying h(x)h(x) into a single elementary function. (b) Verify that your answer in part (a) equals the result obtained by differentiating h(x)h(x) with an appropriate rule and then substituting x=π4x=\tfrac{\pi}{4}. Provide all intermediate steps. (c) Explain why the agreement of the two methods in parts (a) and (b) is mathematically guaranteed for any differentiable function.

Related theory

Browse the problem library