Problems

01

Linear approximation with differentials

Find the linearization of $f(x)=\sqrt[5]{x+1}$ at $a=0$. Then approximate $(0.95{)}^{1/5}$ with your formula.

What is the differential $dx$ you use, and what is the differential $df$ induced by it through the function $f(x)$?

02

Absolute (global) extrema

Find the absolute (global) max and min values of $h(x)$ on the given interval.

• (a) $h(x)=\sqrt[3]{(x-4{)}^2}$ on the interval $x\in [2,5]$
• (b) $h(x)=\ln (x^2+5x+9)$ on the interval $x\in [-3,1]$

03

Analyzing a function for extrema

Consider the function: $$f(x)=\frac{x}{x^2+1}$$

• (a) Compute all critical numbers of $f(x)$.
• (b) On what interval(s) is $f(x)$ decreasing or increasing?
• (c) Use the First Derivative Test to identify the local max and min values and their $x$-coordinates.
• (d) Identify all the inflection points. Where is $f(x)$ concave up and concave down?

04

Sketching a graph - specified properties

Sketch the graph of a function satisfying these properties:

• $f^{\prime }(x)>0$ on $(-\infty ,1)$, $f^{\prime }(x)<0$ on $(1,\infty )$
• $f^{\prime \prime }(x)>0$ on $(-\infty ,-2)$ and on $(2,\infty )$, $f^{\prime \prime }(x)<0$ on $(-2,2)$
• ${\lim }_{x\to -\infty }f(x)=-2$ and ${\lim }_{x\to +\infty }f(x)=0$

05

First derivative test

Compute the derivative of $f(x)=x^{2/3}(6-x{)}^{1/3}$.

Considering your formula for $f^{\prime }$, explain the significance of this chart:

Using the chart, apply the First Derivative Test to the points $0,4,6$ to determine which of these is a local max or min.

06

Function behavior - interpreting graphical features

Consider the following graph of the derivative $f^{\prime }$:

• (a) Describe the intervals where $f$ is increasing or decreasing.
• (b) Find the values of $x$ for which $f$ has a local max or min.
• (c) Where is $f$ concave up or concave down?
• (d) Identify all inflection points of $f$.