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Q 0.

Expert-verifiedFound in: Page 298

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

Problem Zero: Read the section and make your summary of material

the rate can be calculated in terms of the factors that change with time.

"RELATED RATES"

A study of the formula for the surface and area of the right-circumference cylinder, cone, sphere, and rectangular box.

1. A rectangular box

$V=xyz\phantom{\rule{0ex}{0ex}}S=2xy+2yz+2xz$

2. Sphere

$V=\frac{4}{3}\pi {r}^{3}\phantom{\rule{0ex}{0ex}}S=4\pi {r}^{2}$

3. right circular cylinder

$V=\pi {r}^{2}h\phantom{\rule{0ex}{0ex}}S=2\pi rh+2\pi {r}^{2}\phantom{\rule{0ex}{0ex}}L=2\pi rh$4. right circular cone

$V=\frac{1}{3}\pi {r}^{2}h\phantom{\rule{0ex}{0ex}}S=\pi r\sqrt{{r}^{2}+{h}^{2}}+\pi {r}^{2}\phantom{\rule{0ex}{0ex}}L=\pi r\sqrt{{r}^{2}+{h}^{2}}$There are two right triangle theorems.

The Pythagorean principle:${a}^{2}+{b}^{2}={c}^{2}$

2. The law of similar triangles: $\frac{h}{b}=\frac{II}{B},\frac{d}{b}=\frac{D}{B},\frac{d}{h}=\frac{D}{II}$

The first derivative of a function f can be used to calculate the rate of change of a function f when it is increasing or decreasing.

Example: Both the radius and the height can be used to calculate the rate of change in volume of a right circular cylinder. Since the formula $V=\frac{1}{3}\pi {r}^{2}h$ determines the volume. Wherever the radius and height change, the volume V changes as well.

localid="1663925684234" $V\left(t\right)=\frac{1}{3}\pi {\left[r\left(t\right)\right]}^{2}h\left(t\right)\phantom{\rule{0ex}{0ex}}\frac{dV\left(t\right)}{dt}=\frac{1}{3}\pi \left[r(t{)}^{2}\frac{dh\left(t\right)}{dt}+h(t)\frac{dr(t{)}^{2}}{dt}\right]\phantom{\rule{0ex}{0ex}}\frac{dV\left(t\right)}{dt}=\frac{1}{3}\pi \left[r(t{)}^{2}\frac{dh\left(t\right)}{dt}+h(t\left)r\right(t)\frac{dr\left(t\right)}{dt}\right]$

Thus, the rate can be measured in terms of the variables associated over time.

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