26 multiple choice 1 point for values of h very close to 0, which of the following functions best…

26 multiple choice 1 point for values of h very close to 0, which of the following functions best approximates f(x) = (tan(x + h)-tan x)/h? (a) sin x (b) (sin x)/x (c) (tan x)/x (d) sec x (e) sec²x

26 multiple choice 1 point for values of h very close to 0, which of the following functions best approximates f(x) = (tan(x + h)-tan x)/h? (a) sin x (b) (sin x)/x (c) (tan x)/x (d) sec x (e) sec²x

Answer

Answer:

E. $\sec^{2}x$

Explanation:

Step1: Recall the definition of the derivative

The given function $f(x)=\frac{\tan(x + h)-\tan x}{h}$ is in the form of the limit definition of the derivative. The limit as $h\rightarrow0$ of $\frac{f(x + h)-f(x)}{h}$ is $f^\prime(x)$. Here $f(u)=\tan u$.

Step2: Find the derivative of $\tan x$

The derivative of $\tan x$ with respect to $x$ is $\sec^{2}x$. Using the formula $\frac{d}{dx}\tan x=\sec^{2}x$. When $h$ is very close to 0, $\frac{\tan(x + h)-\tan x}{h}\approx\sec^{2}x$.