Math 231 - Essentials of Calculus by James Stewart Prepared by Jason Gaddis Chapter 3 - Applications of Differentiation 3.1 - Maximum and Minimum Values Note We continue our study of functions using derivatives. In this section we find that the derivative can tell us where the maximum and minimum values of a function occur. Definition Let D be the domain of f. A function f has an absolute maximum at c if f(c) f(x) for all x in D. We call f(c) the maximum value of f on D. Similarly, f has an absolute minimum at c if f(c) f(x) for all x in D. We call f(c) the minimum value of f on D. Note The word absolute is sometimes replaced by global Definition A function f has a local maximum at c if f(c) f(x) when x is near c. Similarly, f has a local minimum at c if f(c) f(x) when x is near c Note By near, we mean in an open interval containing c. The word local is sometimes replaced by relative. Example Where are the absolute and local max/min of f(x) = (x 2 1) 4? Theorem (The Extreme Value Theorem) If f is continuous on a closed interval [a, b], then f attains an absolute maximum vale f(c) and an absolute minimum value value f(d) at some numbers c and d in [a, b]. Note If either of the hypotheses above are removed, the theorem is no longer holds. Theorem (Fermat s Theorem) If f has a local maximum or minimum at c, and if f (c) exists, then f (c) = 0. Remark This is only one of many theorems attributed to Pierre de Fermat, who was a lawyer and amateur mathematician. Note The converse to the above theorem is not true in general. Also, there may be more max and min than where this theorem implies (e.g., x ). Definition A critical number of a function f is a number c in the domain of f such that either f (c) = 0 or f (c) does not exist. Example Find the critical numbers of f(x) = x 3 x 2 x. Note As a result of Fermat s Theorem, we can now state: If f has a local max or min at c, then c is a critical number of f. Now we will use what is known as the Closed Interval Method to find absolute max and min values of a continuous function on a closed interval [a, b]. 1. Find the critical numbers of f. 2. Find the values of f at the critical numbers of f in (a, b). 3. Find the values of f at the endpoints of the interval. 4. Choose the largest of all values (in Steps 2 and 3). This is the absolute maximum. 5. Choose the smallest of all values. This is the absolute minimum. Example Find the abs. max and abs. min values of f(x) = x 3 6x 2 + 9x + 2 on the closed interval [ 1, 4]. 1
3.2 - The Mean Value Theorem Theorem (Rolle s Theorem) Let f be a function that satisfies the following three hypotheses: 1. f is continuous on the closed interval [a, b] 2. f is differentiable on the open interval (a, b) 3. f(a) = f(b) Then there is a number c in (a, b) such that f (c) = 0. Example Verify that the function f(x) = x 3 3x 2 + 2x + 5 satisfies the hypotheses of Rolle s Theorem on the interval [0, 2]. Then find all numbers c that satisfy the conclusion of the theorem. Example Show that the equation 2x 1 sin x = 0 has exactly one real root. Theorem (The Mean Value Theorem) Let f be a function that satisfies the following hypotheses: 1. f is continuous on the closed interval [a, b] 2. f is differentiable on the open interval (a, b) Then there is a number c in (a, b) such that Equivalently, f (c) = f(b) f(a). b a f(b) f(a) = f (c)(b a). Example Verify that the function f(x) = x 3 + x 1 satisfies the hypotheses of the Mean Value Theorem on the interval [0, 2]. Then find all numbers c that satisfy the conclusion of the theorem. Theorem If f (x) = 0 for all x in the interval (a, b), then f is constant on (a, b). Corollary If f (x) = g (x) for all x in an interval (a, b), then f g is constant on (a, b); that is, f(x) = g(x)+c where c is a constant. 2
3.3 - Derivatives and the shapes of graphs Note We will see how the derivative gives us information about the behavior of a function. Our first result is a consequence of the Mean Value Theorem. Prop Increasing/Decreasing Test (a) If f (x) > 0 on an interval, then f is increasing on that interval. (b) If f (x) < 0 on an interval, then f is decreasing on that interval. Example Given f(x) = x 4 4x 1, find the intervals on which f is increasing or decreasing. Note Continuing with this idea and the idea of critical numbers, we now have the following result. Prop The First Derivative Test Suppose that c is a critical number of a continuous function f. (a) If f changes from positive to negative at c, then f has a local maximum at c. (b) If f changes from negative to positive at c, then f has a local minimum at c. (c) If f does not change sign at c, then f has no local maximum or minimum at c. Example Find the local minimum and maximum values of the function f in the previous example. Note Now that we have seen what the first derivative can tell us, we turn our attention to the second derivative. Definition If the graph of f lies above all of its tangents on an interval I, then it is is called concave upward (up) on I. If the graph of f lies below all of its tangents on I, it is called concave downard. Definition A point P on a curve y = f(x) is called an inflection point if f is continuous there and the curve switches concavity there. Prop Concavity Test (a) If f (x) > 0 for all x I, then the graph of f is concave upward on I. (b) If f (x) < 0 for all x I, then the graph of f is concave downward on I. Example Find the intervals of concavity and the inflection points of the f in the first example. Prop The Second Derivative Test Suppose f is continuous near c. (a) If f (c) = 0 and f (c) > 0, then f has a local minimum at c. (b) If f (c) = 0 and f (c) < 0, the f has a local maximum at c. Example Given h(x) = (x 2 1) 3, find the following: (a) Intervals of increase or decrease. (b) Local max and min values. (c) Intervals of concavity and inflection points. Sketch the graph of the function using the information above. 3
3.4 - Curve Sketching Note The following is a guide that summarizes much of what you have already learned about the behavior of functions. A. Domain: Before graphing a function, we should decide for what values of x is f(x) defined. B. Intercepts: When it is possible, we should try to determine when the graph crosses the x-axis by letting f(x) = 0 and solving. C. Symmetry: If f( x) = f(x), then the function is even and is symmetric about the y-axis. If f( x) = f(x), then the function is odd and is symmetric about the origin. If f(x + p) = f(x), then the function is periodic and it is sufficient to determine the graph on one period. D. Asymptotes. If the limit as x approaches (or ) is L, then the function has a horizontal asymptote at L. If the limit as x approaches a point (from either side) is infinite, then the graph has a vertical asymptote at that point. E. Intervals of increase or decrease: Use the Increasing/Decreasing Test from 3.3. F. Local max and min values: Find the critical numbers and use the First Derivative Test (alternatively, you may use the Second Derivative Test). G. Concavity and Points of Inflection: Compute f (x) and use the Concavity Test. H. Sketch the curve. Example Graph using the steps above Example Graph using the steps above y = x (x 1) 2. y = sin x tan x. 4
3.5 - Optimization Problems Note We will utilize our techniques for graphing, specifically those skills needed to find absolute minima and maxima, to solve optimization problems. This has great application to areas of buisness and engineering, when we try to make systems as efficient as possible. In these problems, we will want to draw a diagram, when possible, and clearly define our variables. Example Find two numbers whose difference is 100 and whose product is a minimum. function f(x) = x(x 100).) (Hint: use the Example Find the dimensions of a rectangle with area 1000m 2 whose perimeter is as small as possible. Note Sometimes we do not have the luxury of working in a closed interval. Specifically, our domain may be (a, ) for some number a, or (, b) for some number b. For this situation, we have the following modificiation of the First Derivative Test. Prop First Derivative Test for Absolute Extreme Values Suppose that c is a critical number of a continuous function f defined on an interval. (a) If f (x) > 0 for all x < c and f (x) < 0 for all x > c, then f(c) is the absolute maximum value of f. (b) If f (x) < 0 for all x < c and f (x) > 0 for all x > c, then f(c) is the absolute minimum value of f. Example A box with a square base and open top must have a volume of 32000cm 3. Find the dimensions of the box that minimize the amount of material used. 5
3.6 - Newton s Method Note The IVT gave us a way to determine whether a function has a root in a given interval, but it does not (directly) give us a way to find the value of the root. Newton s method provides a method for finding the approximate value of that root. Prop Newton s Method Rather than writing a self-contained formula, we describe the method as follows. Choose a starting value x 1, (sometimes this value is given to expedite the problem). We write the equation of the tangent line at x 1 : y f(x 1 ) = f (x 1 )(x x 1 ). Now we try and find a new value, x 2, that is the root of the previous tangent line. Hence, we let x = x 2 and we solve: 0 f(x 1 ) = f (x 1 )(x 2 x 1 ) x 2 = x 1 f(x 1) f (x 1 ). Now we repeat this process to find x 3 and so on. Then the n t h approximation is x n+1 = x n f(x n) f (x n ). This limit of the sequence (x n ) converges to the root r. Note Newton s method mail fail. This is often the case if f (x 1 ) is too close to zero. In this case, one should choose a better initial approximation. Example Use Newton s method to find the third approximation of x 3 x + 1 = 0 given an initial approximation of x 1 = 1. f( 1) = 1, f (x) = 3x 2 1, f ( 1) = 2. x 2 = x 1 f(x 1) 1 f = 1 (x 1 ) 2 = 1 2 x 3 = x 2 f(x 2) f (x 2 ) Note To approximate within a certain number of decimal places, say eight, we continue until x n and x n+1 agree to eight decimal places. Example Use Newton s method to approximate 3 2 correct to eight decimal places. 6
3.7 - Antiderivatives Definition A function F is called an antiderivative of f on an interval I if F (x) = f(x) for all x in I. Note A function will usually have many antiderivatives. Example Find three antiderivatives of f(x) = x 3. Theorem If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F (x) + C where C is an arbitrary constant. Example Find the most general antiderivative of f(x) = x 3 Example Find the most general antiderivative of f(t) = 3 cos t 4 sin t. Example Find then most general antiderivate of f(x) = x 3/2 Prop Antidifferentiation formulas (see page 186) Definition An equation involving derivatives is a differential equation. Note Using antiderivatives, we can find functions from its derivative if we are given enough information to solve for C. Example Find f given f (x) = 4 6x 40x 3, f(0) = 2, f (0) = 1. Note If a object has position function s(t), then the velocity function is v(t) = s (t). So the position function is an antiderivative of the velocity function. Similarly, the acceleration function is a(t) = v (t), so the velocity function is an antiderivative of the acceleration. If we know s(0) and v(0), then we can find the position function by antidifferentiating twice. Example A particle is moving with the given data. Find the position of the particle: a(t) = 10 + 3t 3t 2, s(0), s(2) = 10. 7