Chapter 6 Notes, Applied Calculus, Tan

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1 Contents 4.1 Applications of the First Derivative Determining the Intervals Where a Function is Increasing or Decreasing Local Extrema (Relative Extrema) Applications of the Second Derivative Determining the Intervals of Concavity Curve Sketching Optimization I Optimization II

2 4.1 Applications of the First Derivative Determining the Intervals Where a Function is Increasing or Decreasing Definition 4.1. Let f(x) be a function defined on an interval I and let x 1 and x 2 be any two point in I. 1. f(x) is increasing on I if x 1 < x 2, then f(x 1 ) < f(x 2 ) 2. f(x) is decreasing on I if x 1 > x 2, then f(x 1 ) < f(x 2 ) The first derivative test for increasing/decreasing. Suppose that f(x) is continuous on [a, b] and differentiable on the open interval (a, b). If f (x) > 0 for all x in (a, b) then f(x) increases on [a, b]. If f (x) < 0 for all x in (a, b) then f(x) decreases on [a, b]. If f (x) = 0 for all x in (a, b) then f(x) is constant on [a, b] Local Extrema (Relative Extrema) Definitions: 1. A function f(x) has a local maximum value at a point c if it is the highest point near itself. 2. A function f(x) has a local minimum value at a point c if it is the lowest point near itself. 3. A critical number of a function f(x) is a number c in the domain of f(x) such that f (x) is zero or f (x) does not exist. The First Derivative Test for Local Extrema. Let f(x) be a continuous function on [a, b] and c be a critical number in [a, b]. 1. If f (x) 0 on (a, c) and f (x) 0 on (c, b), then f(x) has a local maximum of y = f(c) at x = c. 2. If f (x) 0 on (a, c) and f (x) 0 on (c, b), then f(x) has a local minimum of y = f(c) at x = c. 3. If f (x) does not change signs at x = c, then f(x) has no local extrema at x = c. Example Find the intervals where the function is increasing and decreasing. relative extrema for the function. Find the 2

3 Example For each of the following functions find the intervals where the function is icreasing or decreasing then find the relative extrema. 1. f(x) = 2x 2 + 4x + 3 Step 1: Find the critical numbers: Set f (x) = 0 and find where f (x) does not exist Step 2: Make a table 3

4 2. f(x) = x 2 (3 x) 3. f(x) = x 4 4x

5 Example A stone is thrown straight up from the roof of an 80-ft building. The distance (in feet) of the stone from the ground at any time t (in seconds) is given by h(t) = 16t t + 80 When is the stone rising, and when is it falling? If the stone were to miss the building, when would it hit the ground. Sketch the graph of h(t). 5

6 4.2 Applications of the Second Derivative Determining the Intervals of Concavity Definition 4.2. The graph of a differentiable function y = f(x) is concave up on an interval where f (x) is increasing and concave down on an interval where f (x) is decreasing. Question: How do find where f (x) is increasing or decreasing? Answer: The same way we did if for f(x). Take the derivative of f (x) and see where it is positive (increasing) and negative (decreasing). If we take the derivative of the derivative we have found the second derivative. So now we have the second derivative test for concavity: The Second Derivative Test for Concavity Let f(x) be a twice differentiable function on an interval I. 1. If f (x) > 0 on I, the graph of f(x) over I is concave up. 2. If f (x) < 0 on I, the graph of f(x) over I is concave down. Definition 4.3. An inflection point is a point of the graph where the function changes from concave up to concave down or down to up. We can find inflection points with the second derivative: Find where f (x) = 0 and/or where f (x) does not exist Example Find the intervals where the function is concave up/down. Find the inflection points for the function. 6

7 Example Find where f(x) = x 3 + 3x 2 2 is concave up/down. Step 1: Find possible inflection points: f (x) = 0 and/or f (x)does not exist We can also use the second derivative to find maximums and minimums: The Second Derivative Test for Local Extrema Let f(x) be a continuous function on [a, b] and c be a critical point in [a, b]. 1. Compute f (x) and f (x). 2. Find all the critical numbers of f at whcih f (x) = Compute f (c) for each such critical number c. (a) If f (c) < 0, then f(x) has a local maximum of y = f(c) at x = c. (b) If f (c) > 0, then f(x) has a local minimum of y = f(c) at x = c. (c) If f (c) = 0, then the test is inconclusive and you must use the First Derivative Test for Local Extrema. Example For the function f(x) = x 4 + 2x 3 2 find all the extrema using the second derivative test and indicate the intervals where the graph is concave up/down. 7

8 Example For the function f(x) = 1 3 x3 2x 2 5x 10 find all the extrema using the second derivative test and indicate the intervals where the graph is concave up/down. Example Sketch a graph of a function with the following properties. f(2) = 4 f (2) = 0 f (x) > 0 on (, ) 8

9 4.3 Curve Sketching Review A Horizontal Asymptote describes the behavior of a function as x gets very large. Limits at Infinity/Horizontal Asymptotes Let f(x) be the rational function given by f(x) = N(x) D(x) = a nx n + a n 1 x n a 1 x + a 0 b m x m + b m 1 x m b 1 x + b 0 where N(x) and D(x) have no common factors. The lim x f(x) can be determined by comparing the degrees of N(x) and D(x). 1. If n < m, then lim x f(x) = 0 and the graph of f(x) has the line y = 0 (the x-axis) as a horizontal asymptote. 2. If n = m then lim f(x) = a n and the graph of f(x) has the line y = a n x b m b m as a horizontal asymptote. 3. if n > m then lim x f(x) = ± and the graph of f(x) has no horizontal asymptote. Vertical Asymptotes The line x = a is a vertical asyptote of f if either OR lim f(x) = ± x a + lim f(x) = ± x a You find a vertical asymptote of a function f(x) = N(x) by finding a value x = a such D(x) that denominator equals zero D(a) = 0 AND the numerator is not zero N(a) 0. 9

10 Example Find the vertical and horizontal asymptotes for the following functions. 1. f(x) = 1 x f(x) = x 2 3. f(t) = t + 1 2t 1 10

11 The first derivative test for increasing/decreasing. Suppose that f(x) is continuous on [a, b] and differentiable on the open interval (a, b). If f (x) > 0 for all x in (a, b) then f(x) increases on [a, b]. If f (x) < 0 for all x in (a, b) then f(x) decreases on [a, b]. If f (x) = 0 for all x in (a, b) then f(x) is constant on [a, b]. The First Derivative Test for Local Extrema. Let f(x) be a continuous function on [a, b] and c be a critical number in [a, b]. 1. If f (x) 0 on (a, c) and f (x) 0 on (c, b), then f(x) has a local maximum of y = f(c) at x = c. 2. If f (x) 0 on (a, c) and f (x) 0 on (c, b), then f(x) has a local minimum of y = f(c) at x = c. 3. If f (x) does not change signs at x = c, then f(x) has no local extrema at x = c. The Second Derivative Test for Concavity Let f(x) be a twice differentiable function on an interval I. 1. If f (x) > 0 on I, the graph of f(x) over I is concave up. 2. If f (x) < 0 on I, the graph of f(x) over I is concave down. 11

12 Graphing using y and y : Steps: 1. Determine the points of discontinuity. 2. Determine the asymptotes (vertical, horizontal) 3. Determine the x- and y- intercepts. 4. Determine the critical point(s). (Set f (x) = 0 and undefined). 5. Determine the intervals where the function f is increasing/decreasing. 6. Determine the local extrema. 7. Determine the possible point(s) of inflection. Set f (x) = 0 and undefined). 8. Determine the intervals where the function f is concave up/down. 9. Determine the inflection point(s). 10. Determine extra point(s) if necessary. 11. Sketch the graph using the information obtained above. Example Graph f(x) = 1 3 (x 1)3 + 2 using the steps above. 12

13 Example Graph f(x) = 3x 4 + 4x 3 using the steps above. 13

14 Example Graph f(x) = 1 3 (x 1)3 + 2 using the steps above. 14

15 4.4 Optimization I Definition 4.4. Absolute Extrema: The highest or lowest points on the graph. 1. The function f has an absolute maximum value on an interval at a point c if: f(x) f(c) for all x in the interval. The number f(c) is called the maximum value of f on the interval. 2. The function f has an absolute minimum value on an interval at a point c if: f(x) f(c) for all x in the interval. The number f(c) is called the minimum value of f on the interval. 3. The maximum and minimum values of f are called the extreme values of f. NOTE: on an interval extrema can occur at two types of points: a) end points b) critical points Definition 4.5. A critical number of a function f is a number c in the domain of f such that f (c) is zero or f (c) does not exist. To find the extreme values on an interval [a, b]: 1. Find the critical numbers: Set f (x) = 0 and find where f (x) does not exist. 2. Evaluate f at the critical numbers. 3. Evaluate f at the end points x = a and x = b. (ie. f(a) and f(b)) 4. Choose the largest value found in steps 2 and 3 as the maximum value of f on [a, b] and the smallest as the minimum value of f on [a, b]. 15

16 Example Find the local and absolute extrema for f(x) = 2x Step 1: Find the critical numbers: on [0, 5]. Step 2: Evaluate f at the critical numbers. Step 3: Evaluate f at the end points (ie. f(0) and f(5)) Example Find the local and absolute extrema for f(x) = 3x 4 + 4x 3 on [ 2, 1]. 16

17 Example A division of Chapman Corporation manufactures a pager. The weekly fixed cost for the division is $20,000, and the variable cost for producing x pagers/week is dollars. The company realizes a revenue of V (x) = x x x R(x) = 0.02x x (0 x 7500) dollars from the sale of x pagers/week. Find the level of production that will yield a maximum profit for the the manufacturer. 17

18 4.5 Optimization II Word Problems: Steps: 1. Read. 2. Draw a picture when applicable. 3. Determine if you are maximizing or minimizing the problem. 4. Summarize the information in the problem statement. 5. Determine the formula/function that applies. 6. Write the function in terms of one variable. 7. Determine the domain of the function. 8. Determine the critical point(s) 9. Test to determine the extrema. 10. Did you answer the question asked? Example Farmer Bob has 200 ft. of fencing to enclose a rectangular field. What is the largest possible area that he can enclose if he makes 2 side by side corrals by adding a piece of fencing parallel to the shorter side? 18

19 Example Postal regulations specify that a parcel sent by priority mail may have a combined length and girth of no more that 108 in. Find the dimensions of a rectangular package that has a square cross section and the largest volume that may be sent via priority mail. What is the volume of such a package? 19