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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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 The position of an object (in feet) is given by s (t ) = 4 + ln(t 2 ) where t is in seconds. What is the object’s velocity at time t = 3 sec? A. 4 + ln(9) feet/sec B. ln(9) feet/sec C. 2/3 feet/sec D. 1/3 feet/sec E. 1/9 feet/sec
Applications of the Derivative to Science (3/20/09) 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: Physics 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. Linear density: Input: Position on an object such as a wire Output: Mass from the end to that position The derivative is rate of change of mass with respect to position, called the linear density.
Applications: Economics Marginal Cost Input: Some production level Output: The cost of producing at that level The derivative is the rate f change of cost with respect to production level, called the marginal cost. Likewise marginal profit
Related Rates (Monday’s Lab Topic) Suppose two quantities which are related are both changing as time passes (so they are both functions of time). If we start with the equation relating the quantities and take the derivative of both sides with respect to time, then we find out how rates of change are related.
A Simple Example of Related Rates How are the radius of a circle and its area related? Write down an equation. Suppose the circle is growing as time passes. We can find out how the rates of change are related by taking the derivative of both sides with respect to time. Do that. Note that you must use the Chain Rule! If the radius is growing at 3 inches per second, how fast is the area growing at the moment the radius is 20 inches?
Assignment Hand-In #2 is due Thursday at 4:45 pm. Lab #5 on Monday will be in class (not in the computer lab). Please read Section 3.9 in preparation for this lab. Clickers and text are not needed. For Wednesday, read those parts of Section 3.7 which discuss the four science applications on today’s slides, and do Exercises 9, 17, 24 & 29.