Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 2 - Derivatives - 2.2 The Derivative as a Function - 2.2 Exercises - Page 129: 51

Answer

See explanation.

Work Step by Step

\begin{array}{l} \lim _{h \rightarrow 0} \frac{f(h+x)-f(x)}{h} =f(x)\\ =\lim _{h \rightarrow 0} \frac{\left[3(x+h)^{2}+2(x+h)+1\right]-\left(3 x^{2}+2 x+1\right)}{h} \\ =\lim _{h \rightarrow 0} \frac{\left(3 x^{2}+6 x h+3 h^{2}+2 x+2 h+1\right)-\left(3 x^{2}+2 x+1\right)}{h} \\ =\lim _{h \rightarrow 0} \frac{6 x h+3 h^{2}+2 h}{h}=\lim _{h \rightarrow 0} \frac{h(6 x+3 h+2)}{h} \frac{f(x)}{h \rightarrow 0} \frac{\mid i m}{h \rightarrow 0}(6 x+3 h+2)=6 x+2 \\ =\lim _{h \rightarrow 0} \frac{[6(x+h)+2]-(6 x+2)}{h} \\ =\lim _{h \rightarrow 0} \frac{[6 x+6 h+2]-(6 x+2)}{h} \\ =\lim _{h \rightarrow 0} \frac{6 h}{h}=\lim _{h \rightarrow 0} 6=6 \end{array}
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