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Understanding Derivatives: Applications in Sciences and Problem Solving

This content delves into the concept of derivatives in various fields such as biology, economics, and physics, illustrating their significance through practical applications like population growth, marginal costs, and motion examples. It includes two clicker questions focusing on the derivatives of specific functions and the slope of tangent lines. The material also discusses how derivatives are employed to determine rates of change and includes assignments to reinforce understanding of these concepts through targeted exercises.

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Understanding Derivatives: Applications in Sciences and Problem Solving

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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 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

  3. 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.

  4. 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

  5. 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?

  6. 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!)

  7. 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.

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