Consider the function f(x)=x2e6xf(x)=x2e6x. For this function there are three important intervals: (??,A](-?,A], [A,B][A,B], and [B,?)[B,?) where AA and BB are the critical numbers. Find AA and BB For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC). (??,A](-?,A]: [A,B][A,B]: [B,?)[B,?)

License Question 1. Points possible: 2 This is attempt 1 of 1.

Evaluate the limit using L’Hospital’s rule limx?0ex+3x?14xlimx?0ex+3x-14x

License Question 2. Points possible: 2 This is attempt 1 of 1.

Given the function g(x)=4×3?36×2+96xg(x)=4×3-36×2+96x, find the first derivative, g'(x)g?(x). g'(x)=g?(x)= Notice that g'(x)=0g?(x)=0 when x=2x=2, that is, g'(2)=0g?(2)=0. Now, we want to know whether there is a local minimum or local maximum at x=2x=2, so we will use the second derivative test. Find the second derivative, g”(x)(x). g”(x)=(x)= Evaluate g”(2)(2). g”(2)=(2)= Based on the sign of this number, does this mean the graph of g(x)g(x) is concave up or concave down at x=2x=2? [Answer either up or down — watch your spelling!!] At x=2x=2 the graph of g(x)g(x) is concave Based on the concavity of g(x)g(x) at x=2x=2, does this mean that there is a local minimum or local maximum at x=2x=2? [Answer either minimum or maximum — watch your spelling!!] At x=2x=2 there is a local

License Question 3. Points possible: 2 This is attempt 1 of 1.

Let f(x)=x4?18x27f(x)=x4-18×27. (a) Use the definition of a derivative or the derivative rules to find f'(x)=f?(x)= (b) Use the definition of a derivative or the derivative rules to find f”(x)=f??(x)= (c) On what interval is ff increasing (include the endpoints in the interval)? interval of increasing = (d) On what interval is ff decreasing (include the endpoints in the interval)? interval of increasing = (e) On what interval is ff concave downward (include the endpoints in the interval)? interval of increasing = (f) On what interval is ff concave upward (include the endpoints in the interval)? interval of increasing =

License Question 4. Points possible: 2 This is attempt 1 of 1.

Below is the function f(x)f(x). 1234567-1-2-3-4-5-6-71234567-1-2-3-4-5-6-7 Over which interval of xx values is f’>0f?>0?

· (?2,?)(-2,?)

· [?2,?)[-2,?)

· (??,?2)(-?,-2)

· (??,?2](-?,-2]

· (??,?](-?,?]

Over which interval of xx values is f’<0f?<0? · (?2,?)(-2,?) · [?2,?)[-2,?) · (??,?2)(-?,-2) · (??,?2](-?,-2] · (??,?](-?,?] Over the interval (??,?)(-?,?), this function is · concave up (f''>0f??>0)

· concave down (f”<0f??<0) Get help: Video License Question 5. Points possible: 2 This is attempt 1 of 1. Consider the function f(x)=?5x2+10x?2f(x)=-5x2+10x-2. f(x)f(x) has a critical point at x=x= . At the critical point, does f(x)f(x) have a local min, a local max, or neither? License Question 6. Points possible: 2 This is attempt 1 of 1. Mark the critical points on the following graph. 12345-1-2-3-4-561218243036-6-12-18 Clear All Draw: Dot License Question 7. Points possible: 2 This is attempt 1 of 1. Consider the function f(x)=12x5+45x4?200x3+2f(x)=12x5+45x4-200x3+2. For this function there are four important intervals: (??,A](-?,A], [A,B][A,B],[B,C][B,C], and [C,?)[C,?) where AA, BB, and CC are the critical numbers. Find AA and BB and CC At each critical number AA, BB, and CC does f(x)f(x) have a local min, a local max, or neither? Type in your answer as LMIN, LMAX, or NEITHER. At AA At BB At CC License Question 8. Points possible: 2 This is attempt 1 of 1. A fence 2 feet tall runs parallel to a tall building at a distance of 6 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? License Question 9. Points possible: 2 This is attempt 1 of 1. Evaluate the limit using L'Hospital's rule limx?09x?11xxlimx?09x-11xx License Question 10. Points possible: 2 This is attempt 1 of 1. 1234-1-2-3-4123-1-2-3 For the above rational function f( x ) = 11x4x2+1.511x4x2+1.5, identify its three inflection points. lowest = middle = highest = License Question 11. Points possible: 2 This is attempt 1 of 1. 123-1-2-312345678 For the above rational function f( x ) = 154x2+2154x2+2, identify the interval on which f is concave down. < x < License Question 12. Points possible: 2 This is attempt 1 of 1. Consider the function f(x)=32e?x22f(x)=32e-x22. f(x)f(x) has two inflection points at x = C and x = D with C?DC?D where CC is and DD is Finally for each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD). (??,C](-?,C]: [C,D][C,D]: [D,?)[D,?) License Question 13. Points possible: 2 This is attempt 1 of 1. Consider the function f(x)=3x+93x+1f(x)=3x+93x+1. For this function there are two important intervals: (??,A)(-?,A) and (A,?)(A,?) where the function is not defined at AA. Find AA For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC). (??,A)(-?,A): (A,?)(A,?) Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD). (??,A)(-?,A): (A,?)(A,?) License Question 14. Points possible: 2 This is attempt 1 of 1. Let f(x)=x4?72x211f(x)=x4-72x211. (a) Use the definition of a derivative or the derivative rules to find f'(x)=f?(x)= (b) Use the definition of a derivative or the derivative rules to find f''(x)=f??(x)= (c) On what interval is ff increasing (include the endpoints in the interval)? interval of increasing = (d) On what interval is ff decreasing (include the endpoints in the interval)? interval of increasing = (e) On what interval is ff concave downward (include the endpoints in the interval)? interval of increasing = (f) On what interval is ff concave upward (include the endpoints in the interval)? interval of increasing = License Question 15. Points possible: 2 This is attempt 1 of 1. Consider the function f(x)=e?(x?17)218f(x)=e-(x-17)218. f(x)f(x) has two inflection points at x = C and x = D with C?DC?D where CC is and DD is Finally for each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD). (??,C](-?,C]: [C,D][C,D]: [D,?)[D,?) License Question 16. Points possible: 2 This is attempt 1 of 1. A box with a square base and open top must have a volume of 340736 cm3cm3. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only xx, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of xx.] Simplify your formula as much as possible. A(x)=A(x)= Now, calculate when the function A(x)A(x) has a minimum. The length of the side of the square bottom is The minimum amount of material needed is License Question 17. Points possible: 2 This is attempt 1 of 1. Consider the function f(x)=5x3?6xf(x)=5x3-6x on the interval [?2,2][-2,2]. Find the average or mean slope of the function on this interval. By the Mean Value Theorem, we know there exists at least one cc in the open interval (?2,2)(-2,2) such that f'(c)f?(c) is equal to this mean slope. For this problem, there are two values of cc that work. The smaller one is and the larger one is Get help: Video License Question 18. Points possible: 2 This is attempt 1 of 1. Let f(x)=x3+3x2?105x+4f(x)=x3+3x2-105x+4. (a) Use the definition of a derivative or the derivative rules to find f'(x)=f?(x)= (b) Use the definition of a derivative or the derivative rules to find f''(x)=f??(x)= (c) On what interval is ff increasing (include the endpoints in the interval)? interval of increasing = (d) On what interval is ff decreasing (include the endpoints in the interval)? interval of decreasing = (e) On what interval is ff concave downward (include the endpoints in the interval)? interval of downward concavity = (f) On what interval is ff concave upward (include the endpoints in the interval)? interval of upward concavity = License Question 19. Points possible: 2 This is attempt 1 of 1. 12-1-2-3-4-5-17-34-51-68-85-102-119-136-153 For the above quartic polynomial f( x ) = x42+2x3?9x2x42+2x3-9x2, identify the interval on which f is concave down. < x < License Question 20. Points possible: 2 This is attempt 1 of 1. Consider the function f(x)=2x+2x?1f(x)=2x+2x-1. For this function there are four important intervals: (??,A](-?,A], [A,B)[A,B),(B,C](B,C], and [C,?)[C,?) where AA, and CC are the critical numbers and the function is not defined at BB. Find AA and BB and CC For each of the following intervals, tell whether f(x)f(x) is increasing (type in INC) or decreasing (type in DEC). (??,A](-?,A]: [A,B)[A,B): (B,C](B,C]: [C,?)[C,?) Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(x)f(x) is concave up (type in CU) or concave down (type in CD). (??,B)(-?,B): (B,?)(B,?): License Question 21. Points possible: 2 This is attempt 1 of 1. Consider the function f(x)=2?x+7f(x)=2x+7 on the interval [3,5][3,5]. Find the average or mean slope of the function on this interval. By the Mean Value Theorem, we know there exists a cc in the open interval (3,5)(3,5) such that f'(c)f?(c) is equal to this mean slope. For this problem, there is only one cc that works. Find it. License Question 22. Points possible: 2 This is attempt 1 of 1. A box with a square base and open top must have a volume of 97556 cm3cm3. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only xx, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of xx.] Simplify your formula as much as possible. A(x)=A(x)= Next, find the derivative, A'(x)A?(x). A'(x)=A?(x)= Now, calculate when the derivative equals zero, that is, when A'(x)=0A?(x)=0. [Hint: multiply both sides by x2x2.] A'(x)=0A?(x)=0 when x=x= We next have to make sure that this value of xx gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x)(x). A"(x)=(x)= Evaluate A"(x)(x) at the xx-value you gave above. NOTE: Since your last answer is positive, this means that the graph of A(x)A(x) is concave up around that value, so the zero of A'(x)A?(x) must indicate a local minimum for A(x)A(x). (Your boss is happy now.) License Question 23. Points possible: 2 This is attempt 1 of 1. The function f(x)=?2x3+36x2?162x+10f(x)=-2x3+36x2-162x+10 has one local minimum and one local maximum. This function has a local minimum at xx = with value and a local maximum at xx = with value License Question 24. Points possible: 2 This is attempt 1 of 1. A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 250 cubic centimeters of soup. The sides and bottom of the container will be made of syrofoam costing 0.04 cents per square centimeter. The top will be made of glued paper, costing 0.07 cents per square centimeter. Find the dimensions for the package that will minimize production cost. Helpful information: h : height of cylinder, r : radius of cylinder Volume of a cylinder: V=?r2hV=?r2h Area of the sides: A=2?rhA=2?rh Area of the top/bottom: A=?r2A=?r2 To minimize the cost of the package: Radius: cm Height: cm Minimum cost: cents License Question 25. Points possible: 2 This is attempt 1 of 1. The function f(x)=2x3?30x2+126x+4f(x)=2x3-30x2+126x+4 has one local minimum and one local maximum. This function has a local minimum at xx equals with value and a local maximum at xx equals with value License Question 26. Points possible: 2 This is attempt 1 of 1. A baseball team plays in a stadium that holds 66000 spectators. With the ticket price at $8 the average attendence has been 27000. When the price dropped to $6, the average attendence rose to 33000. Assume that attendence is linearly related to ticket price. What ticket price would maximize revenue? $ License Question 27. Points possible: 2 This is attempt 1 of 1. Mark the critical points on the following graph. x1e?x22x1e-x22, 0.2 12-1-20.20.40.60.81-0.2-0.4-0.6-0.8-1 Clear All Draw: Dot License Question 28. Points possible: 2 This is attempt 1 of 1. Evaluate the limit using L'Hôpital's rule. limx??15xe1x?15xlimx??15xe1x-15x License Question 29. Points possible: 2 This is attempt 1 of 1. Find the critical numbers of the function f(x)=4x5?15x4?20x3+8f(x)=4x5-15x4-20x3+8 and classify them. x = is a x = is a x = is a License Question 30. Points possible: 2 This is attempt 1 of 1. Let f(x)=x3+6x2?135x+20f(x)=x3+6x2-135x+20. (a) Use the definition of a derivative or the derivative rules to find f'(x)=f?(x)= (b) Use the definition of a derivative or the derivative rules to find f''(x)=f??(x)= (c) On what interval is ff increasing (include the endpoints in the interval)? interval of increasing = (d) On what interval is ff decreasing (include the endpoints in the interval)? interval of decreasing = (e) On what interval is ff concave downward (include the endpoints in the interval)? interval of downward concavity = (f) On what interval is ff concave upward (include the endpoints in the interval)? interval of upward concavity =