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 /

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2.6 The Derivative By Dr. Julia Arnold

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Presentation transcript:

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 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 (2/7/11) 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 Wednesday Read pages 221 through 223 of Section 3.7 up to Example 2. Do Exercises 1 a.b.c.g., 3 a.b.c.g., 7 and 9 on pages