Question

1. evaluate the following derivatives on your calculator. be sure to interpret your calculator’s answer correctly. (a) $f(pi)$, if $f(x) = \tan x$ (b) $left. \frac{dy}{dx} \

ight|_{x=3}$ if $y = \frac{|2x - 6|}{4}$ (c) $y(2.3)$ if $y = \sqrt{x + 1}$

Explanation:

Step1: Differentiate $f(x)=\tan x$

$f'(x)=\sec^2 x = \frac{1}{\cos^2 x}$

Step2: Evaluate at $x=\pi$

$\cos(\pi)=-1$, so $f'(\pi)=\frac{1}{(-1)^2}=1$

Step1: Simplify $y=\frac{|2x-6|}{4}$ for $x>3$

For $x>3$, $2x-6>0$, so $y=\frac{2x-6}{4}=\frac{x-3}{2}$

Step2: Differentiate and evaluate at $x=3$

$y'=\frac{1}{2}$. The right-hand derivative at $x=3$ is $\frac{1}{2}$; note: the left-hand derivative is $-\frac{1}{2}$, so the derivative does not exist at $x=3$ in the strict sense, but the calculator will return the right-hand limit value $\frac{1}{2}$.

Step1: Rewrite $y=\sqrt{x+1}$

$y=(x+1)^{\frac{1}{2}}$

Step2: Differentiate using power rule

$y'=\frac{1}{2}(x+1)^{-\frac{1}{2}}=\frac{1}{2\sqrt{x+1}}$

Step3: Evaluate at $x=2.3$

$y'(2.3)=\frac{1}{2\sqrt{2.3+1}}=\frac{1}{2\sqrt{3.3}} \approx \frac{1}{2\times1.8166} \approx 0.275$

Answer:

(a) $1$
(b) The derivative does not exist at $x=3$ (right-hand derivative is $\frac{1}{2}$)
(c) $\approx 0.275$ (exact value: $\frac{1}{2\sqrt{3.3}}$)