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

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clicker question 1
Clicker Question 1
  • What is the derivative of f (x ) = arctan(5x )?
    • A. arcsec 2 (5x )
    • B. 5 arcsec 2 (5x )
    • C. 5 / (1 + 5x2)
    • D. 5 / (1 + 25x2)
    • E. 1 / (1 + 25x2)
clicker question 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
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
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
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
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
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 230-231.