MATH 151, Fall 2015, Week 12, Section 5.1-5.3 Chapter 5 Application of Differentiation We develop applications of differentiation to study behaviors of functions and graphs Part I of Section 5.1-5.3, Qualitative/intuitive approach Terminology f(x) is said to be increasing on an open interval (a, b) if x 1 < x 2 f(x 1 ) < f(x 2 ) f(x) is said to be decreasing on an open interval (a, b) if x 1 < x 2 f(x 1 ) > f(x 2 ) f(x) has a local maximum (local minimum) at x = a if f(a) is the largest (smallest) value on some interval containing a. (Local minimum = relative minimum) A curve is said to be concave upward (concave downward) if it bends downward. An Inflection point is the point where a curve changes the concavity. Example 1. A graph of a function f(x), defined on [0, 6], is shown below. f on f on f is concave upward on f is concave downward on Inflection point of f is 1
Example 2. Let f(x) = 1 12 x4 x 2 + 2.5. f (x) = f (x) = f on f on f (x) > 0 on f (x) < 0 on Loc. max of f(x) is at Loc. min of f(x) is at f (x) changes its sign at f is concave upward on f is concave downward on f (x) > 0 on f (x) < 0 on Inflection point of f is f (x) changes its sign at Thm f on (a, b) f is concave UP on (a, b) f on (a, b) f is concave DOWN on (a, b) Inflection point The 1 st Derivative Test If f changes its sign from + to at x = c then f has a loc. at c. If f changes its sign from to + at x = c then f has a loc. at c. 2
The 2 nd Derivative Test If f (c) = 0 and f (c) > 0, then f has a loc at c. If f (c) = 0 and f (c) < 0, then f has a loc at c. Example 3. Check the above test with f(x) = x 3 3x and g(x) = (x 2) 3. Ex 3. The graph of the second derivative f of a function f is shown. Find the inflection points of f. Ex 4. Use the graph to state the absolute and local maximum and minimum values of the function. 3
Part II of Section 5.1-5.3, Quantitative approach f(x) has an absolute maximum (or absolute minimum) at x = a if f(a) is the largest (smallest) value on the domain. The maximum and minimum values are extreme values. A critical number of f(x) is a number c in the domain such that f (c) = 0 or f (x) does not exists. Absolute maximum= global maximum, absolute minimum= global minimum. Ex. 5 Find the absolute and local maximum and minimum values of f. (If an answer does not { exist, enter DNE.) 36 x 2 if 6 x < 0 f(x) = 4x 1 if 0 x 6 The extreme value thm If f is a continuous function on a closed interval [a, b], then f has an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d on [a, b]. The closed interval method To find the absolute max/min of a continuous function f on a closed interval [a, b]: Step 1. Find critical numbers of f, Step 2. Find f(c) for critical numbers of f Step 3. Find f(a) and f(b). Step 4. The largest value in Step 2 and Step 3 is the absolute max; the smallest value is the absolute min. 4
Ex 6. Find absolute maximum and minimum for the following functons (1) f(x) = 4x 3 18x 2 48x + 5 on [ 2, 5] (2) f(t) = 4 cos t + 2 sin 2t on [0, π/2] (Use cos 2t = 1 2 sin 2t) (3) f(x) = xe x2 /5 on [ 3, 10] Mean Value Theorem (MVT) Suppose f(x) is continuous on an interval [a, b] and differentiable on (a, b), then there exists a number c, where a < c < b, so that f (c) = f(b) f(a) b a Graphically, this means the tangent line to the graph of f(x) at x = c is parallel to the secant line joining two end points (a, f(a)) and (b, f(b)). Ex 7. Does f(x) = x satisfy the hypotheses of the MVT on [1, 8]? Find all numbers x + 4 c that satisfy the conclusion of the MVT. 5
Table method for Graph Sketch Step 1 Find f and f, and then solve f (x) = 0 and f (x) = 0. Step 2 Draw a 4-line table with x values from step 1 Step 3 Determine signs(±) of f and f to tell the behavior of f (by plugging in). Step 4 Fill the bottom row with numbers and curves for f(x), and then sketch the graph of f. Ex 8. Plot the graph of f(x) = 1 12 x4 x 2 + 2.5 6
Ex 8-1. Use the table method to plot the graph of f(x) = (4 x)e x. Ex 8-2. Use the table method to plot the graph of f(x) = x 4 ln x. 7
Ex 9. Suppose the function f(x) has the first derivative f (x) = (a) Find the interval(s) where f(x) is increasing/decreasing. (b) Find the x-coordinates of all local extrema on the graph of f(x). 2(x 2)e8x (x + 3) 3, x 3. Ex 9. Suppose the function g(x) has the second derivative g (x) = (x 1) 5 (x+4)(x 10) 8. (a) Find the interval(s) where f(x) is concave up/down. (b) Find the x-coordinates of inflection points. 8
Section 5.5 Applied Max/Min Problems Finding extreme values have practical applications in many areas of life. Step 1 Read the problem carefully and then draw a diagram. Step 2 Introduce symbols(letters), and establish a function f for those. Step 3 Eliminate variables to express f as a function of x. Step 4 Apply closed interval method to find Abs. Max/Min. Ex 10. Find two numbers whose sum is 100 and whose product is maximum. Ex 10. Find two positive numbers whose product is 100 and whose sum is minimum. Ex 11. A farmer wants to fence an area of 6 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence? 9
Ex 12. You have 100 feet of fencing to construct a pen with four equal sized stalls as shown. What are the dimensions of the pen of largest area and what is the largest area? Ex 13. If 10, 800cm 2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box. Ex 14.A piece of wire 10m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How much wire should be used for the square in order to minimize the total area? 10