Clicker Question 1

What is the derivative of f (x ) = arctan(5x )? A. arcsec 2 (5x ) B. 5 arcsec 2 (5x ) C. 5 / (1 + 5x 2) D. 5 / (1 + 25x 2) E. 1 / (1 + 25x 2)

Clicker Question 2

What is the derivative of g (x ) = ln(arctan(x ))? A. 1 / (1 + x 2) B. 1 / ((1 + x 2) arctan(x )) C. 1 / arctan(x ) D. 2x / ((1 + x 2) arctan(x )) E. 2x / arctan(x )

Clicker Question 3

What is the slope of the tangent line to the curve y = x arcsin(x) at the point (1, /2)? A. 0 B. 1 C. /2 D. ½ E. undefined

Applications of the Derivative to the Sciences (2/3/12)

Sciences (both natural and social) have numerous applications of the derivative. Some examples are:

Population growth or decay (Biology etc.) Input: time Output: the size of some population The derivative is the rate of growth or decay

of that population with respect to time.

Applications: Economics

Marginal Cost Input: Some production level Output: The cost of producing at that

level The derivative is the rate of change of

cost with respect to production level, called the marginal cost.

Likewise marginal profit

Applications: Physics There are many such applications.

We look at just one easy one: Velocity:

Input: time Output: position of a moving object The derivative is the rate of change of

position with respect to time, i.e., velocity. The second derivative is the rate at which

the velocity is changing. What’s that called?

Example of Velocity & Acceleration

Suppose the position of a car on a highway (in miles from the start) is given by s(t) = 50t + 3 sin(t ) where t is in hours. What is its position after 5 hours? What is its velocity after 5 hours? What is its acceleration after 5 hours?

Include units in all answers!

Assignment for Monday Read pages 224 through 226 of

Section 3.7 up to Example 2. Do Exercises 1 a.-g., 3 a.-g. and 9

on pages 233-234.