APPLICATIONS OF THE DERIVATIVE CHAPTER 3 APPLICATIONS OF THE DERIVATIVE

3.1 Maxima & Minima Maxima: point whose function value is greater than or equal to function value of any other point in the interval Minima: point whose function value is less than or equal to function value of any other point in the interval Extrema: Either a maxima or a minima

Where do extrema occur? Peaks or valleys (either on a smooth curve, or at a cusp or corner) f’(c)=0 or f’(c) is undefined Discontinuties Endpoints of an interval These are known as the critical points of the function Once you know you have a critical point, you can test a point on either side to determine if it’s a max or min (or maybe neither…just a leveling off point)

3.2 Monotonicity and Concavity

Let f be defined on an interval I (open, closed, or neither). Then f is INCREASING on I if, DECREASING on I if, MONTONIC on I if it is ether increasing or decreasing

Monotonicity Theorem Let f be continuous on an interval I and differentiable at every interior point of I. If f’(x)>0 for all x interior to I, then f is increasing on I If f’(x)<0 for all x interior to I, then f is decreasing on I.

Concave UP vs. Concave DOWN Let f be differentiable on an open interval I. If f’ is increasing on I, f is concave up (the graph appears to be curved up, as a container that would hold water) If f’ is decreasing on I, f is concave down (the graph appears to be curved down, as if a container were dumping water out)

Point of Inflection Where concavity changes: goes from concave up to concave down (or vice versa) f’ is neither increasing or decreasing, the change in f’ = 0, thus f’’=0

Find inflection points & determine concavity for f(x) Inflection pts: x=-2,0,1 Concave up: (-2,0), (1,infinity) Concave dn: (-inf.,-2), (0,1)

3.3 Local Extrema and Extrema on Open Intervals

First Derivative Test Let f be continuous on an open interval (a,b) that contains a critical point c. If f’(x)>0 for all x in (1,c) and f’(x)<0 for all x in (c,b), then f(c) is a local max. value. If f’(x)<0 for all x in (1,c) and f’(x)>0 for all x in (c,b), then f(c) is a local min. value. If f’(x) has the same sign on both sides of c, then f(c) is not a local extreme value.

Second Derivative Test Let f’ and f’’ exist at every point in an open interval (a,b) containing c, and suppose that f’(c)=0. a) If f’’(c)<0, then f© is a local max. value of f. b) If f’’(c)>0, then f© is a local min. value of f.

3.4 Practical Problems Optimization problems – finding the “best” or “least” of “most cost effective”, etc. often involves finding the extrema of the function Use either 1st or 2nd derivative test

Example A fence is to be constructed using three lengths of fence (the 4th side of the enclosure will be the side of the barn). I have 120 yd. of fencing and the barn is 150’ long. In order to enclose the largest possible area, what dimensions of fence should be used? (continued on next slide)

Example continued Area is to be optimized: A = l x w Perimeter = 120 yd = 360’=2l + w w = 360’ – 2l So, 2 lengths of 90’ and a width of 180’. HOWEVER, the barn is only 150’ wide, so in order to enclose the greatest area, we won’t use a critical point of the function, rather we will evaluate the area using the endpoints of the interval, with w=150’. The length = 55’ and the area enclosed = 8250 sq. ft.

3.5 Graphing Functions Using Calculus

Critical points & Inflections points If f’(x) = 0, function levels off at that point (check on either side or use 2nd deriv. test to see if max. or min.) If f’(x) is undefined: cusp, corner, discontinuity, or vertical asymptote (look at behavior and limits of function on either side) If f’’(x) = 0: inflection point, curvature changes

3.6 Mean Value Theorem for Derivatives If f is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there is at least one number c in (a,b) where

Example: Find a point within the interval (2,5) where the instantaneous velocity is the same as the average velocity between t=2 and t=5.

If functions have the same derivatives, they differ by a constant. If F’(x) = G’(x) for all x in (a,b), then there is a constant C such that F(x) = G(x) + C for all x in (a,b).

3.7 Solving Equations Numerically Bisection Method Newton’s Method Fixed-Point Algorithm

Bisection Method Let f(x) be a continuous function, and let a and b be numbers satisfying a<b and f(a) x f(b) < 0. Let E denote the desired bound for the error (difference between the actual root and the average of a and b). Repeat steps until the solution is within the desired bound for error. Continue next slide.

Bisection Method

Newton’s Method Let f(x) be a differentiable function and let x(1) be an initial approximation to the root r of f(x) = 0. Let E denote a bound for the error. Repeat the following step for n = 1,2,… until the difference between successive error terms is within the error.

Fixed-Point Algorithm Let g(x) be a continuous function and let x(1) be an initial approximation to the root ro of x = g(x). Let E denote a bound for the error (difference between r and the approximation). Repeat the following step for n – 1,2,… until the difference between succesive approximations are within the error.

3.8 Antiderivatives Definition: We call F an antiderivative of f on an interval if F’(x) = f(x) for all x in the interval.

Power Rule

Integrate = Antidifferentiate Indefinite integral = Antiderivative Constants can be moved out of the integral Integral of a sum is the sum of the integrals Integral of a difference is the difference of the integrals

3.9 Introduction to Differential Equations An equation in which the unknown is a function and that involves derivates (or differentials) of this unknown function is called a differential equation. We will work with only first-order separable differential equations.

Example Solve the differential equation and find the solution for which y = 3 when x = 1.

Example continued If x = 1 and y = 3, solve for C.