미분 전체 문제

Choose $x$ to minimize the sum of their areas.

For the following exercises, consider two nonnegative numbers $x$ and $y$ such that $x+y=10.$ Maximize and minimize the quantities.

$xy$

For the following exercises, consider two nonnegative numbers $x$ and $y$ such that $x+y=10.$ Maximize and minimize the quantities.

${x}^{2}{y}^{2}$

For the following exercises, consider two nonnegative numbers $x$ and $y$ such that $x+y=10.$ Maximize and minimize the quantities.

$y-\frac{1}{x}$

For the following exercises, consider two nonnegative numbers $x$ and $y$ such that $x+y=10.$ Maximize and minimize the quantities.

${x}^{2}-y$

Find the volume of the largest right circular cylinder that fits in a sphere of radius $1.$

Find the volume of the largest right cone that fits in a sphere of radius $1.$

Find the area of the largest rectangle that fits into the triangle with sides $x=0,y=0$ and $\frac{x}{4}+\frac{y}{6}=1.$

Find the largest volume of a cylinder that fits into a cone that has base radius $R$ and height $h.$

Find the dimensions of the closed cylinder volume $V=16\pi$ that has the least amount of surface area.

Find the dimensions of a right cone with surface area $S=4\pi$ that has the largest volume.

For the following exercises, consider the points on the given graphs. Use a calculator to graph the functions.

[T] Where is the line $y=5-2x$ closest to point $(1,1)?$

For the following exercises, consider the points on the given graphs. Use a calculator to graph the functions.

[T] Where is the parabola $y={x}^{2}$ closest to point $(2,0)?$

For the following exercises, consider the points on the given graphs. Use a calculator to graph the functions.

[T] Where is the parabola $y={x}^{2}$ closest to point $(0,3)?$

A window is composed of a semicircle placed on top of a rectangle. If you have $20\,\text{ft}$ of window-framing materials for the outer frame, what is the maximum size of the window you can create? Use $r$ to represent the radius of the semicircle.

You have a garden row of $20$ watermelon plants that produce an average of $30$ watermelons apiece. For any additional watermelon plants planted, the output per watermelon plant drops by one watermelon. How many extra watermelon plants should you plant?

You are building five identical pens adjacent to each other with a total area of $1000{\,\text{m}}^{2},$ as shown in the following figure. What dimensions should you use to minimize the amount of fencing?

If $f(-1)=-6$ and $f(1)=2,$ then there exists at least one point $x\in [-1,1]$ such that ${f}^{'}(x)=4.$

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample. Assume that $f(x)$ is continuous and differentiable unless stated otherwise.

If ${f}^{'}(c)=0,$ there is a maximum or minimum at $x=c.$

There is a function such that $f(x)<0,{f}^{'}(x)>0,$ and ${f}^{''}(x)<0.$ (A graphical “proof” is acceptable for this answer.)