230 likes | 470 Views
Application of Derivatives. Jake Albert, Hardik Joshi, Hyun Kim. 1 st and 2 nd derivative tests. 1 st Derivative Test Used to find relative max/min at critical points Find derivative and determine critical values ‘Line Test’ for critical points. Line Test. Rolle’s Theorem.
E N D
Application of Derivatives Jake Albert, Hardik Joshi, Hyun Kim
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
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!!!!
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?????
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
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
2nd Derivative Test Simple Sample Critical Value at
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)
Linearization Just for thought—is this estimate higher or lower than the actual value ?
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
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?
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
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”
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
Thanks to this site for their wonderful derivative graphs • http://www.math.montana.edu/frankw/ccp/calculus/deriv/compare/answer.htm#answer1