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Clicker Question 1

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

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Clicker Question 1

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  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)

  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 )

  3. 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

  4. 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.

  5. 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

  6. 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?

  7. 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!

  8. 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.

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