Use Linear approximation to estimate a function

Question

Use linear approximation to estimate $\tan(\frac{\pi}{4} + 0.05)$. Identify the differentials $dy$ and $dx$ in the situation.

I know that the form for linear approximation is $f'(x)*dx$. I calculated the derivative of this function to be $0$, so I am not sure what to do when it comes to identifying $dy$ and $dx$.

Answer 1

This

the form for linear approximation is $f′(x)∗dx$

is too vague to be useful.

The definition of the derivative says that when $h$ is small, $$ \frac{f(x+h) - f(x)}{h} \approx f'(x). $$

Multiply that approximation by $h$ and rearrange to get $$ f(x+h) \approx f(x) + h \times f'(x). $$ Now cleverly choose $f$, $x$ and $h$ to get the approximation you want.

If you must use differentials (I wouldn't) then $dx$ is $h$, the small change in $x$, and $dy$ is the corresponding small change in $f(x)$.

Answer 2

Approximate as follows,

$$dy=f'(x)dx$$

or, specifically,

$$\tan\left(\frac{\pi}{4} + 0.05\right)-\tan\left(\frac{\pi}{4}\right)=\left[\tan\left(\frac{\pi}{4}\right)\right]'\cdot 0.05$$

Thus, the approximation is,

$$\tan\left(\frac{\pi}{4}+0.05\right) = 1+\sec^2\left(\frac\pi4\right)\cdot0.05=1+2\cdot0.05=1.1$$

Note that the derivative $\sec^2(x)$ is evaluated at $\frac\pi4$, which is not zero.