5. Graphical Applications of Derivatives

Find the exact length of the curve $y=\frac{x^3}{3}+\frac{1}{4x}$ for $1\le x\le 3$.

Find the length of the loop of the curve given by $x=6t-2t^3$, $y=6t^2$.

Use symmetry to evaluate the double integral of $f(x,y)=xy$ over the region $R$, where $R$ is the disk $x^2+y^2\le 4$.

Which of the following best describes the end behavior of the function $f\left(x\right)=2x^3-3x^2-3$?

Given the function $f\left(x\right)=x^3-12x$, on which interval is the function increasing?

For the function graphed below, which interval contains a local maximum? $[-1,2]$, $[2,4]$, $[4,6]$, or $[6,8]$?

Suppose the graph of $f\left(x\right)$ is shown below. At which of the following intervals is $f^{″}\left(x\right)$ $>0$?

Suppose $f\left(x\right)$ is continuous on the interval $[0,4]$ and $f\left(x\right)>0$ for all $x$ in $(1,3)$. Which of the following could be the entire interval over which $f\left(x\right)$ is positive?

Let $f(x,y)=x^2+3y^2$. Find the maximum rate of change of $f$ at the point $(1,2)$ and the direction in which it occurs.

For the function $f\left(x\right)$ defined on the interval $[0,6]$, the graph shows a peak at $x=2$ and a trough at $x=5$. Which interval contains the local maximum of $f\left(x\right)$?

Find the global maximum and minimum values of the function $f(x,y)=e^{-xy}$ on the region defined by $x^2+4y^2\le 1$.

Let $f\left(x\right)=x^3-3x^2+1$. What is the absolute minimum value of $f$ on the closed interval $[0,3]$?

What is the end behavior of the graph of the polynomial function $y=10x^9-4x$?

For the curve $y=1+40x^3-3x^5$, at which point does the tangent line have the largest slope?

Let $f\left(x\right)=x^3+a{x}^2$. For what value of $a$ does $f\left(x\right)$ have a critical point at $x=1$?