Exercise 1.1.35 (35-38)

Question

Evaluate the difference quotient for the given function. Simplify your answer.
    \begin{enumerate}
        \item [35.] $f(x)=4+3x-x^2,\qquad \frac{f(3+h)-f(3)}{h}$
        \item [36.] $f(x)=x^3, \qquad \frac{f(a+h)-f(a)}{h}$
        \item [37.] $f(x)=\frac{1}{x}, \qquad \frac{f(x)-f(a)}{x-a}$
        \item [38.] $f(x)=\sqrt{x+2}, \qquad \frac{f(x)-f(1)}{x-1}$
    \end{enumerate}

Khagan Gulusoy · Accepted Answer

\begin{enumerate}

\item[35.]
$$
\begin{aligned}
\dfrac{f(3+h)-f(3)}{h}& =\dfrac{4+3(3+h)-(3+h)^2-\left(4+3 \cdot 3-3^2ight)}{h} \
& =\dfrac{4+9+3 h-9-6 h-h^2-4-9+9}{h} \
& =\dfrac{-h^2-3 h}{h}=-h-3 .
\end{aligned}
$$

\item[36.]

$$
\begin{aligned}
\dfrac{f(a+h)-f(a)}{h} & =\dfrac{(a+h)^3-a^3}{h}=\frac{a^3+3 a^2 h+3 a h^2+h^3-a^3}{h} \
& =\dfrac{3 a^2 h+3 a h^2+h^3}{h}=3 a^2+3 a h+h^2 .
\end{aligned}
$$
\item[37.]

$$
\begin{aligned}
 \dfrac{f(x)-f(a)}{x-a}&=\dfrac{\frac{1}{x}-\dfrac{1}{a}}{x-a}
=\dfrac{\frac{a-x}{x a}}{x-a}=\dfrac{a-x}{(x-a) x a} \
& =-\dfrac{1}{x a} 	ext {. } 
\end{aligned}
$$
\item[38.]
$$
\begin{aligned}
\dfrac{f(x)-f(1)}{x-1}&=\dfrac{\sqrt{x+2}-\sqrt{1+2}}{x-1}=\dfrac{\sqrt{x+2}-\sqrt{3}}{x-1} \
& =\dfrac{\sqrt{x+2}-\sqrt{3}}{(\sqrt{x+2}-\sqrt{3})(\sqrt{x+2}+\sqrt{3})}=\dfrac{1}{\sqrt{x+2}+\sqrt{3}} . 
\end{aligned}
$$
\end{enumerate}

Exercise 1.1.35 (35-38)

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Exercise 1.1.35 (35-38)

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