Q 0. - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Given information

"RELATED RATES"

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

Step 2: Explanation

1. A rectangular box

$$\begin{gathered} V=x y z \\ S=2 x y+2 y z+2 x z \end{gathered}$$

2. Sphere

$$\begin{gathered} V=\frac{4}{3}\pi r^3 \\ S=4\pi r^2 \end{gathered}$$

3. right circular cylinder

$$\begin{gathered} V=\pi r^{2}h \\ S=2\pi rh+2\pi r^2 \\ L=2\pi rh \end{gathered}$$

4. right circular cone

$$\begin{gathered} V=\frac{1}{3}\pi r^{2}h \\ S=\pi r\sqrt{r^2+h^2}+\pi r^2 \\ L=\pi r\sqrt{r^2+h^2} \end{gathered}$$

Step 3: Further calculation

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.

$$\begin{gathered} V\left(t\right)=\frac{1}{3}\pi {\left[r\left(t\right)\right]}^{2}h\left(t\right) \\ \frac{dV\left(t\right)}{dt}=\frac{1}{3}\pi \left[r(t{)}^2\frac{dh\left(t\right)}{dt}+h(t)\frac{dr(t{)}^2}{dt}\right] \\ \frac{dV\left(t\right)}{dt}=\frac{1}{3}\pi \left[r(t{)}^2\frac{dh\left(t\right)}{dt}+h(t\left)r\right(t)\frac{dr\left(t\right)}{dt}\right] \end{gathered}$$

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