Application of Derivatives

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.

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Application of Derivatives

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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
Line Test
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``
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

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