Calculus (I)

Calculus (I)

W^EN-C^HING L^IEN

Department of Mathematics National Cheng Kung University

2008

Ch3-4: Applications

Example(1)

Suppose that we pump water into an inverted

right-circular conical tank at the rate of 5 cubic feet per minute.

The tank has a height of 6 ft and the radius on top is 3 ft.

How fast is the water level rising when the water is 2 ft deep?

(V = ¹πr²h)

Example(2)

Assume that the radius r and the volume V = ⁴₃πr³of a sphere are differentiable function of t .

Express dV

dt in terms of dr dt

§ Implicit Differentiation

Ex1: Find dy dx if x²

4 + y² 9 =1.

§ Higher Derivatives

Ex2: Fine f⁽²⁾(x),f⁽³⁾(x)for f(x) =√

x f(x) = x

x+1 f(x) =xⁿ

§ Implicit Differentiation

Ex1: Find dy dx if x²

4 + y² 9 =1.

§ Higher Derivatives

Ex2: Fine f⁽²⁾(x),f⁽³⁾(x)for f(x) =√

x f(x) = x

x+1 f(x) =xⁿ

§ Implicit Differentiation

Ex1: Find dy dx if x²

4 + y² 9 =1.

§ Higher Derivatives

Ex2: Fine f⁽²⁾(x),f⁽³⁾(x)for f(x) =√

x f(x) = x

x+1 f(x) =xⁿ

§ Implicit Differentiation

Ex1: Find dy dx if x²

4 + y² 9 =1.

§ Higher Derivatives

Ex2: Fine f⁽²⁾(x),f⁽³⁾(x)for f(x) =√

x f(x) = x

x+1 f(x) =xⁿ

§ Implicit Differentiation

Ex1: Find dy dx if x²

4 + y² 9 =1.

§ Higher Derivatives

Ex2: Fine f⁽²⁾(x),f⁽³⁾(x)for f(x) =√

x f(x) = x

x+1 f(x) =xⁿ

Thank you.