Section 4.4 Enhanced Assignment

Student: Jalen Wilson Date: 07/12/24 Instructor: Neelam Aggarwal Course: Spring 2022 - Calculus I (MATH1441-02F) Assignment: Section 4.4 Enhanced Assignment 1. The function V(x) = ​x(14 − 2​x)(15 − ​x), 0 < x < 7​, models the volume of a box. (​ a) Find the extreme values of V. ​(b) Interpret any values found in part ​(a) in terms of the volume of the box. (​ a) The extreme​value(s) of V​is/are 288 ​(Use a comma to separate answers as​needed.) . ​(b) Select the correct choice below and fill in the answer box within the choice. ​(Use a comma to separate answers as​needed.) A. The value found in the previous step represents the smallest volume of the box and occurs when x = . B. The values found in the previous step represent the smallest and the largest volumes of the box and they occur when x = and x = ​, respectively. C. The value found in the previous step represents the largest volume of the box and occurs when x = 3 . 2. Identify the inflection points and local maxima and minima of the function graphed to the right. Identify the open intervals on which the function is differentiable and is concave up and concave down. Find the inflection​point(s). Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. y 3 y= x 3 2 − 2x − 5x x A. The​point(s) is/are 2, − 46 3 . (​ Type an ordered​pair, using integers or fractions. Simplify your answer. Use a comma to separate answers as​needed.) B. There are no inflection points. Find each local maximum. Select the correct choice below​and, if​necessary, fill in the answer​box(es) to complete your choice. A. There is one local maximum value of 8 3 at x = −1 . (​ Simplify your answers. Type integers or​fractions.) B. There are two local maxima. In increasing order of​x-value, the values are and at x = and x = ​, respectively. (​ Simplify your answers. Type integers or​fractions.) C. There are no local maxima. Find each local minimum. Select the correct choice below​and, if​necessary, fill in the answer​box(es) to complete your choice. A. There is one local minimum value of − 100 3 at x = 5 . (​ Simplify your answers. Type integers or​fractions.) B. There are two local minima. In increasing order of​x-value, the values are and at x = and x = ​, respectively. ​(Simplify your answers. Type integers or​fractions.) C. There are no local minima. Find the open​interval(s) on which the function is differentiable and concave up. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The​interval(s) is/are (2,∞) . (​ Simplify your answer. Use a comma to separate answers as needed. Type your answer in interval notation.) B. The function is never concave up B. The function is never concave up. Find the open​interval(s) on which the function is differentiable and concave down. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The​interval(s) is/are ( − ∞,2) . (​ Simplify your answer. Use a comma to separate answers as needed. Type your answer in interval notation.) B. The function is never concave down. 3. Identify the coordinates of any local and absolute extreme points 2 and inflection points. Graph the function y = x − 8x + 12. Identify the coordinates of any local maximum points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The local maximum​point(s) is/are . ​(Type an ordered pair. Use a comma to separate answers as​needed.) B. There are no local maximum points. Identify the coordinates of any local minimum points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The local minimum​point(s) is/are (4, − 4) . (​ Type an ordered pair. Use a comma to separate answers as​needed.) B. There are no local minimum points. Identify the coordinates of the absolute maximum point. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The absolute maximum point is . ​(Type an ordered​pair.) B. There is no absolute maximum point. Identify the coordinates of the absolute minimum point. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The absolute minimum point is (4, − 4) . ​(Type an ordered​pair.) B. There is no absolute minimum point. Identify the inflection​point(s). Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The​point(s) is/are . ​(Type an ordered pair. Use a comma to separate answers as​needed.) B. There are no inflection points. Use the graphing tool to graph the function. 10 y 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 x 2 4 6 8 10 4. Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. 3 2 y = − 2x + 6x − 8 Identify the coordinates of the local maximum points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The local maximum​point(s) is/are (2,0) . (​ Type an ordered pair. Use a comma to separate answers as​needed.) B. There are no local maximum points. Identify the coordinates of the local minimum points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The local minimum​point(s) is/are (0, − 8) . (​ Type an ordered pair. Use a comma to separate answers as​needed.) B. There are no local minimum points. Identify the coordinates of the absolute maximum points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The absolute maximum​point(s) is/are . ​(Type an ordered pair. Use a comma to separate answers as​needed.) B. There are no absolute maximum points. Identify the coordinates of the absolute minimum points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The absolute minimum​point(s) is/are . ​(Type an ordered pair. Use a comma to separate answers as​needed.) B. There are no absolute minimum points. Find the inflection points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The inflection​point(s) is/are (1, − 4) . (​ Type an ordered pair. Use a comma to separate answers as​needed.) B. There are no inflection points. Use the graphing tool to graph the function. 10 8 6 4 2 -10 -8 -6 -4 -2-2 -4 -6 -8 -10 y x 2 4 6 8 10 5. Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y = 7x + 7 sin x​, 0 ≤ x ≤ 2π Identify the coordinates of the local maximum points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The local maximum​point(s) is/are (2π,14π) . (​ Type an ordered pair. Type an exact answer in terms of π. Use a comma to separate answers as needed. Do not use commas in the individual​coordinates.) B. There are no local maximum points. Identify the coordinates of the local minimum points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The local minimum​point(s) is/are (0,0) . (​ Type an ordered pair. Type an exact answer in terms of π. Use a comma to separate answers as needed. Do not use commas in the individual​coordinates.) B. There are no local minimum points. Identify the coordinates of the absolute maximum points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. (2π,14π) A. The absolute maximum​point(s) is/are . (​ Type an ordered pair. Type an exact answer in terms of π. Use a comma to separate answers as needed. Do not use commas in the individual​coordinates.) B. There are no absolute maximum points. Identify the coordinates of the absolute minimum points. Select the correct choice below​and, if necessary, fill in the answer box to complete your choice. A. The absolute minimum​point(s) is/are (0,0) . (​ Type an ordered pair. Type an exact answer in terms of π. Use a comma to separate answers as needed. Do not use commas in the individual​coordinates.) B. There are no absolute minimum points. Identify the coordinates of any inflection points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. (π,7π) A. The function has an inflection point at . (​ Type an ordered pair. Type an exact answer in terms of π. Use a comma to separate answers as needed. Do not use commas in the individual​coordinates.) B. The function has no inflection points. Choose the correct graph of y = 7x + 7 sin x. A. y 28 C. y y 56 14 0 B. 56 28 x 0 π 2π 0 D. 28 x 0 π 2π 0 y 56 28 x 0 π 2π 0 x 0 π 2π 6. Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. y= 8x 2 x + 36 Identify the coordinates of the local maximum points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The local maximum​point(s) is/are 6, 2 . 3 (​ Type an ordered pair. Type an exact​answer, using radicals as needed. Use a comma to separate answers as​needed.) B. There are no local maximum points. Identify the coordinates of the local minimum points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The local minimum​point(s) is/are − 6, − 2 . 3 (​ Type an ordered pair. Type an exact​answer, using radicals as needed. Use a comma to separate answers as​needed.) B. There are no local minimum points. Identify the coordinates of the absolute maximum points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The absolute maximum​point(s) is/are 6, 2 . 3 (​ Type an ordered pair. Type an exact​answer, using radicals as needed. Use a comma to separate answers as​needed.) B. There are no absolute maximum points. Identify the coordinates of the absolute minimum points. Select the correct choice below​and, if necessary, fill in the answer box to complete your choice. A. The absolute minimum​point(s) is/are − 6, − 2 3 . (​ Type an ordered pair. Type an exact​answer, using radicals as needed. Use a comma to separate answers as​needed.) B. There are no absolute minimum points. Find the inflection points. Select the correct choice below​and, if​necessary, fill in the answer box to complete your choice. A. The inflection​point(s) is/are −6 3,− 1 3 , 6 3, 1 3 ,(0,0) . (​ Type an ordered pair. Type an exact​answer, using radicals as needed. Use a comma to separate answers as​needed.) B. There are no inflection points. Now sketch the graph. Choose the correct graph below. A. B. C. D. 9 y 3 y 3 x y 9 x y x x 7. The first derivative of a continuous function y = ​f(x) is given. Find y′′ and then sketch the general shape of the graph of f. y′ = 4 sin x​, − π 2 ≤x≤ 3π 2 The second derivative is y′′ = 4 cos x . Choose the correct sketch of the general shape of the graph of f. A. B. 5 y C. 5 5 x − 3π 3π 2 2 D. y y x −π -5 2π x − 3π 3π 2 2 -5 6 x −π -5 2π -5 8. The given figure shows the graphs of the first and second derivatives of a function y = ​f(x). Sketch the approximate graph of​f, given that the graph passes through the point P. y = f′(x) y 5 Which of the following is the correct graph of the function y = ​f(x)? A. B. y y y 6 P 6 P P x -6 x 6 -6 -6 6 -6 x -6 6 C. D. y y 6 P 6 P x y = f′′(x) -6 -6 x 6 -6 -6 -6 9. Graph the following rational function. 2 y= − x −4 2 x −2 Choose the correct graph below. A. B. 8 y C. 8 y x -8 8 -8 D. 8 y 8 x -8 8 -8 y x -8 8 -8 6 x -8 8 -8 10. Graph the rational function shown below. y= 5x 2 x −4 Choose the correct graph below. A. B. 10 y C. 10 x -10 y D. 10 y x 10 -10 -10 10 10 x -10 -10 10 -10 y x -10 10 -10 3 2 11. Find the values of constants​a, b, and c so that the graph of y = ax + bx + cx has a local maximum at x = 2​, local minimum at x = − 4​, and inflection point at ​( − 1​, − 26​). a= −1 b= −3 c= 24 ​(Simplify your answers. Type integers or simplified​fractions.) 12. Use the graph of the function​f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down. 10 y B. Local minimum at x = ​1; local maximum at x = − ​1; concave down on ​(0,∞); concave up on ​( − ∞​,0) 8 6 4 2 -10 -8 -6 -4 -2 x 2 -2 A. Local minimum at x = ​1; local maximum at x = − ​1; concave up on ​(0,∞​); concave down on ​( − ∞​,0) 4 6 8 10 -4 C. Local maximum at x = ​1; local minimum at x = − ​1; concave up on ​( − ∞​,∞​) D. Local maximum at x = ​1; local minimum at x = − ​1; concave up on ​(0,∞​); concave down on ​( − ∞​,0) -6 -8 -10 13. Use the graph of the function​f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down. 10 y B. Local maximum at x = ​3; local minimum at x = − ​3; concave up on ​(0, − ​3) and ​(3,∞​); concave down on ​( − ​3,3) 8 6 4 2 -10 -8 -6 -4 -2 -2 -4 -6 -8 -10 A. Local minimum at x = ​3; local maximum at x = − ​3; concave up on ​(0, − ​3) and ​(3,∞​); concave down on ​( − ​3,3) x 2 4 6 8 10 C. Local minimum at x = ​3; local maximum at x = − ​3; concave up on ​(0,∞​); concave down on ​( − ∞​,0) D. Local minimum at x = ​3; local maximum at x = − ​3; concave down on ​(0,∞); concave up on ​( − ∞​,0) 14. Graph the rational function. y= x−3 2 x −9 A. B. 6 -6 6 6 -6 C. -6 6 6 -6 D. -6 6 6 -6 -6 6 -6