Application of Derivatives

1. Application of Derivatives Jake Albert, Hardik Joshi, Hyun Kim

2. 1st and 2nd derivative tests • 1st Derivative Test • Used to find relative max/min at critical points • Find derivative and determine critical values • ‘Line Test’ for critical points

3. Line Test

4. Rolle’s Theorem • Rolle’s Theorem • If f(x) is continuous on [a,b], differentiable on (a,b) and f(a)=f(b) • THEN a ‘c’ value exists between a and b, such that f`(c)=0 • In fact, in this one there are TWO of them!!!!

5. Mean Value Theorem • If f(x) is continuous on [a,b] • THEN a ‘c’ exists between a and b such that f`(c)=this • Remember—where does this expression COME from?????

6. Concavity and POI • Concavity relates to the change in the slope of a function • At Point of Inflection, f``(x) is _________________________???? equal to 0, or is Undefined

7. 2nd Derivative Test • Used to determine local extrema of a function • If f(x) is continuous, and “c” is a critical value (that is, f`(x) is ) • THEN if f`(x) exists and • f``>0, then • f``<0, then • f``=0, then we have an inconclusive test equal to 0, or is Undefined f(c) is a local min f(c) is a local max

8. 2nd Derivative Test Simple Sample Critical Value at

9. Linear Approximation • Linear Approximation • Linearization equation will have the same slope is the approximated value “a” used for finding the derivative Estimate. The slope of our linearized function is , going through the point (16,2)

10. Linearization Just for thought—is this estimate higher or lower than the actual value ?

11. Particle Motion • If the position function is given as a function of t , or s(t) • v(t), or the velocity function, is equal to s’(t) • a(t), or the acceleration function, is equal to v’(t) or s’’(t) • How would you find the average rate of change over a certain time interval???? The change in position divided by the change in time for that interval

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14. Optimization • Remember to sketch to get an idea of what the question is asking. • Write the target equation you wish to optimize • Write constraint equations that limit various variables • Combine the equations What dimensions provide a maximum volume for a rectangular solid with a square base, if the surface area is 150 square milimeters?

15. Optimization (cont.) • Determine the “feasible domain” of the equation in the combined equation • Find the first derivative of the combined equation and determine the critical points (in the hopes of finding a relative maximum or minimum within the feasible domain) • ANSWER THE QUESTION We know x must be greater than 0 Critical values at 5,-5, but -5 does not lie within the feasible domain. Dimensions are 5 mm x 5 mm x 5 mm

16. Related Rates A hot-air balloon rises straight upwards from a flat field and is tracked 500 feet away from lift-off point. At the moment the range finder’s elevation angle is the angle is increasing at the rate of 0.14 radians/minute. How fast is the balloon rising? • Identify any variables needed in the problem (sketch if felt needed) • Use equations that relate the variables in the problem. • Differentiate the equations with respect to t, the time variable • Substitute all known values to solve for the answer the question is asking for • REMEMBER: If you are asked “how fast” something changes, you give a positive answer, while either a negative or positive answer suits the question of “what rate”

17. Related Rates Example (cont.) Implicit differentiation of the equation relating the variables  Solving for the change in height  Answer the question!! The Balloon is rising at 280 feet per minute

18. Thanks to this site for their wonderful derivative graphs • http://www.math.montana.edu/frankw/ccp/calculus/deriv/compare/answer.htm#answer1