Math 140 Written Homework 5: 2.8, 3.1, 3.2 Page 1 of 7From the questions below, please solve ALL 10 problems.The homework is worth 10 points. Each question is worth 1 points.Show all of your work and put a box around your final answer.Number each attempted question clearly.Write legibly (that is, suitably large and suitably dark); if the grader can’t read your answer, it’s consideruncompleted.Question 1 For f(x) = x3e2x+1, write the limit definition of f0(4). Don’t simplify or evaluate the limit. Forexample, if g(x) = x3you would writeg0(1) = limh→0(1 + h)3− 13hand that’s it.Question 2 Suppose that f is a function given byf(x) =|x + 5| if x ≤ −1x1/5if − 1 < x ≤ 322x − 62 if x > 32.(a) At which x-values does f not have a limit?(b) At which x-values is f not continuous?(c) At which x-values is f not differentiable?Math 140 Written Homework 5: 2.8, 3.1, 3.2 Page 2 of 7Question 3 Some people who don’t appreciate the finer things in life brew coffee in a percolator which results insubpar coffee, since the balance of compounds is not ideal when extraction occurs at the boiling pointof water. As a connoisseur, you prefer to brew your coffee at 93◦C. You pour over your coffee at t = 0,where t is in minutes, eventually transfer it to your insulated mug, and finish drinking it after onehour. The temperature of your coffee T = f (t), in degrees Celsius, at various times is given below.t 0 10 20 30 40 50 60T = f (t) 93 74 59 49 44 42 41Assume that f(t) is a decreasing function on its domain 0 ≤ t ≤ 60.(a) Estimate f0(20), and include units.(b) Write a sentence explaining the practical interpretation of the statement f (35) = 47.(c) Write a sentence explaining the practical interpretation of the statement f−1(43) = 44.(d) Write a sentence explaining the practical interpretation of the statement f0(10) = −1.7.Math 140 Written Homework 5: 2.8, 3.1, 3.2 Page 3 of 7Question 4 For the given graph of the function f , on the left, sketch the graph of of f0on the right.−4 −2 2 4−4−224−4 −2 2 4−4−224Question 5 For the given graph of the function f , on the left, sketch the graph of of f0on the right.−4 −2 2 4−4−224−4 −2 2 4−4−224Question 6 For the given graph of the function f , on the left, sketch the graph of of f0on the right.−4 −2 2 4−4−224−4 −2 2 4−4−224Math 140 Written Homework 5: 2.8, 3.1, 3.2 Page 4 of 7Question 7 The spread of a virus is modeled byV (t) = −t3+ t2+ 12t,where V (t) is the number of people (in hundreds) with the virus and t is the number of weeks sincethe first case was observed.(a) Sketch V (t).(b) What is a reasonable domain of t for this problem?(c) Find the average rate of infection from t = 0 to t = 2.(d) Find the instantaneous rate of infection as a function of t using the limit definition of the derivative.(e) Find V (2) and V0(2). Write a sentence interpreting V (2) and V0(2) in terms of the number ofinfected people. Make sure to include units.(f) Sketch the tangent line to the graph you drew in a. at the point (2, V (2)). State the slope of thetangent line.(g) Use V (2) and V0(2) to estimate the value of V (2.1).(h) Find the maximum number of people infected at the same time and when the maximum occurs.Determine the rate of infection at this time.Math 140 Written Homework 5: 2.8, 3.1, 3.2 Page 5 of 7Question 8 Let P (t) be a deer population (in thousands) of an ecological region t years after 2000. A graph of thederivative P0(t) is give below. Use it to answer the following questions.2 4 6 8 10 12 14−2246tP0(a) Calculate P0(8) and interpret that in terms of the deer population.(b) Indicate the years during which the deer population declined.(c) Order the following from least to greatest. If there is a tie, indicate that with an equals sign. Youdo not need to find the values themselves: P0(1), P0(3), P0(6), P0(7), P0(9), P0(10).(d) Order the following from least to greatest. If there is a tie, indicate that with an equals sign. Youdo not need to find the values themselves: P (0), P (2), P (3), P (5), P (7), P (9).(e) Order the following from least to greatest. If there is a tie, indicate that with an equals sign. Youdo not need to find the values themselves: P (9), P (11), P (14), P (15).(f) Evaluate the limit limh→0P0(2 + h) − P0(2)h.Math 140 Written Homework 5: 2.8, 3.1, 3.2 Page 6 of 7Question 9 From the above graph of y = f (x), find all x-value(s) at which f is not differentiable.Math 140 Written Homework 5: 2.8, 3.1, 3.2 Page 7 of 7Question 10 Use the Intermediate Value Theorem to show that there is a solution of the equation, x3− 4x = 1 in(−1,