Review 4 Extreme Values Points of Inflection Justifying Pulling info from a chart Mapping f, f, f Tying it all together How do you determine when a function has a max? The first derivative changes from positive to negative for a max. You find this change by setting the first derivative = 0 (show this!!) and solving. You may graph the derivative on a calculator and see where it crosses the x axis. Non Calculator question chart it for sign changes Apr 8 7:44 AM Apr 8 10:05 AM How do you determine when a function has a minimum? The first derivative changes from negative to positive. You find the change by setting the derivative = 0 (show this!!) or graph the derivative on a calculator and find where it crosses the x axis. Non calculator question chart it for sign changes. To justify you must talk about the sign of the derivative and how it relates to the slope of the function. f >0, therefore f increases f <0, therefore f decreases. f changes from + to at x = a, therefore f(a) (the y value) is a local maximum. You may mention x = a after, or only if the original function is not known. Otherwise, give the entire point as the max. **if the function is NOT called f, make sure that you talk about the function with the correct name! Apr 8 10:05 AM Apr 8 10:15 AM The graph of g: explain the derivative of g Function find it s derivative and explain it through a graph. Function Derivative Explaining the derivative from g: g is increasing from ( 4, A) therefore g is positive. g is decreasing from (A, B) therefore g is negative. g is increasing from (b, 4) therefore g is positive. THIS IS THE ONLY TIME WE TALK ABOUT g INC IN RELATION TO g' BEING POSITIVE. NORMALLY, WE GO THE OTHER DIRECTION. The derivative has zeros at Apr 8 10:16 AM Apr 8 10:17 AM 1
Function find the first derivative and explain it by charting Charting to explain the derivative in relationship to the function g zeros are at How do you determine when a function is concave up? When the second derivative is positive. Translation: Apr 8 10:19 AM Apr 8 10:20 AM How do you determine if a function is concave down? When the second derivative is negative. To justify f you must talk about the second derivative in relation to the first derivative s slope or in relation to the actual function. f >0, therefore f increases or f is concave up. f < 0, therefore f decreases or f is concave down. You must talk about the correct function s name! If you cannot remember what name was given for the function, then just talk about the second derivative, first derivative, and call the function the function! Apr 8 10:21 AM Apr 8 10:22 AM The graph of the second derivative of g Function..finding the second derivative and explaining. Given Pay Attention To What You Are Given In The Question!! Function, First Derivative, Second Derivative. Talk about the derivative of g to explain g or g. g is positive from ( 4, C) therefore g is increasing or g is concave up. g is negative from (C, D) therefore g is decreasing or g is concave down. g is positive from (D, 4) therefore g is increasing or g is concave up. If this is a calculator question, graph the second derivative and find where the graph crosses the x axis. If not, chart it. Apr 8 10:22 AM Apr 8 10:23 AM 2
Chart for previous problem How do you determine if the particle is speeding up or slowing down? The signs for the first and second derivative (velocity and acceleration) are the same for a specific location the particle is speeding up. v(b) and a(b) are both positive or negative The signs are opposite, the particle is slowing down. They like to ask this from time to time. Apr 8 10:24 AM Apr 8 10:25 AM Mapping f to f Graph of f Describe the derivative graph based off of the graph of f. When the function increasing then the slopes are positive, then the derivative is graphed above the x axis. When the function is decreasing then the slopes that are negative, then the derivative is graphed below the x axis. f decreases from ( 4, A) therefore f is negative (graphed below the x axis). f increases from (A, 4), therefore f is positive (graphed above the x axis) Apr 8 10:27 AM Apr 8 10:27 AM Mapping f to f When the first derivative is below the x axis (negative y's), then the function is decreasing. When the first derivative is above the x axis (positive y's), then the function is increasing. When the derivative crosses the x axis, then the function has a max or a min. If the derivative touches the x axis, then it is a critical point candidate only. Graph of f Describe the graph of f from f' f' < 0 from (, A), therefore f is decreasing. f' > 0 from (A, B), therefore f is increasing. f' <0 from (B, ), therefore f is decreasing. There is a max at x = B since f' changes from + to, and there is a min at c = A since f' changes from to +. (No y's since we only have the graph of f') Apr 8 10:28 AM Apr 8 10:28 AM 3
Mapping f to f If you have the graph of the first derivative and want to know about concavity, then the "slopes" (increasing and decreasing) behavior of the derivative graph tells you about the second derivative. When the first derivative increases, then f'' >0, therefore the function is concave up. When the first derivative decreases, then f'' <0, therefore the function is concave down. Graph of f to f Describe the graph of f'' from the graph of f'. f' is increasing from (, ), therefore f'' is positive (always above the x axis). The graph of f'' will be a horizontal line. Apr 8 10:29 AM Apr 8 10:30 AM Mapping f to f When you have the graph of the second derivative, the graph tells you about the slopes of the first derivative and where the concavity of the function changes. When f'' is below the x axis, f' is decreasing and the function is concave down. When f'' is above the x axis, f' is increasing and the function is concave up. Remember Max/min are always a y value. If you have the ability to find the y coordinate, you must find it and describe the max/min as the y You mention the x after stating that the max/min is (y value) So the max is at x = Or the max occurs at (x, y) Apr 8 10:30 AM Apr 8 10:31 AM Remember The only time you do not mention the y is if you are given the first derivative (graph or equation) and NOT the function This is the only time you just talk about the x in respect to the max/min since there is no way to find the y (you don't have the original function or it's constant "c"). If the information in the problem states x coordinate or x value, then just talk about x. Justifying explaining all parts Let f be the function given by A) find the and You are looking at the right and left of the graph. This could be a calculator or a non calculator question. Do you know the behavior of the graph as you go to the left? Apr 8 10:31 AM Apr 8 10:32 AM 4
Do you know the behavior of the graph as you go to the left? If you evaluate negative numbers in an exponent, it relocates the part with the exponent to the denominator. The numerator will be negative, while the denominator grows without bound. So the limit is 0. Explanation: Do you know the behavior of the graph as you go to the right? As you evaluate numbers to the right, the entire function grows without bound, so there is no limit. Explanation is: bound or grows without Apr 8 10:33 AM Apr 8 10:34 AM Part B B) Find the absolute minimum value of f. Justify that your answer is an absolute minimum. Step 1 take the derivative and set it equal to zero Then solve for the x s and explain the signs of the derivative Finally, state why the y coordinate is the minimum value Solve for 1+2x Chart around ½ Explain the chart Apr 8 10:34 AM Apr 8 10:34 AM Explain Part C f'(x) is negative on (, 1/2) therefore f(x) decreases. f'(x) is positive on ( 1/, ), therefore f(x) increases. f'(x) changes from negative to positive at x = 1/2. Therefore the absolute minimum is 1/e. Evaluate in f f( 1/2) is 1/e or appox. 0.368 with a calculator. Apr 8 10:36 AM Apr 8 10:37 AM 5
Part D The abstract question D) Consider the family of functions defined by, where b is a nonzero constant. Show that the absolute minimum value of is the same for all nonzero values of b. What you do Take the derivative and set it equal to zero Remember that "b" is a constant so treat it like one. Why use y'? It was defined that way for part d. Factor out the common factor Set equal to zero and solve Evaluate in the original function x = 1/b, so in original fuction it becomes y = 1/e So y has an absolute minimum value of 1/e for all nonzero b Apr 8 10:38 AM Apr 8 10:39 AM 6