Exercises

01

Linear approximation - practice

Find the linear approximation of each function near the given basepoint. Use your formula to compute the requested approximation, and then compare to the exact value by giving the error (the difference from the exact value).

• (a) For $f(x)=x^4$, near basepoint $a=2$. Approximate $1.99^4$ and give the error.
• (b) For $f(x)=\sqrt[3]{x}$ and $a=1$. Approximate $\sqrt[3]{0.94}$ and give the error.

02

Linear approximation - differential notation

Suppose $f(x)=x^2-4x$. Work at $x=2$.

• (a) What is $\Delta y$ (the exact delta), given $\Delta x=0.5$?
• (b) What is $dy$ (the approximated delta), given the use of $\Delta x=0.5$ for $dx$?

03

Critical numbers - practice

For each function below, find the largest possible domain, and the set of all critical numbers.

• (a) $f(x)=(x+1{)}^2(x-3)$
• (b) $g(x)=\frac{4x}{x^2+16}$
• (c) $h(x)=\frac{\sqrt[3]{x}}{x-1}$