Math 140 Written Homework 6: 3.3, 3.4, 3.5, 3.6, 3.7 Page 1 of 2From the questions below, please solve 5 problems only.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 Calculate the derivative indicated using the given function.(a) f00(1), f(t) =t1 + t(b)d3xdt3, x(t) = t−3/4Question 2 Find the derivative of each function.(a) f(θ) = sec θ + 3θ(b) f(z) =14zez−116ez(c) g(y) =cos yy2sin y + 3ey(d) h(x) =tan xx3+ exQuestion 3 Suppose that the function f(x) and f0(x) have the following values at x = 0 and x = 1.x f(x) f0(x)0 9 −21 −3 1/5Find the first derivatives of the following combination at the given value of x.(a)f(x)x + cos(x), x = 0(b) ex(f(x))2, x = 1Question 4 Use the quotient rule to derive the formula forddxcsc x and forddxcot x.Question 5 Some values of a differentiable, invertible function f and its derivative are given below.x 0 2 4 6 8f(x) 15 13 10 6 2f0(x) −1 −2 −2.5 −3 −5Calculate the following.(a) g0(2), where g(x) = f (x)e3x.(b) h0(6), where h(x) = xef(x).Question 6 Use the table below to evaluate the derivatives below.x f(x) f0(x) g(x) g0(x) h(x) h0(x)1 2 3 4 5 -2 -1(a) Let F (x) = f(x)g(x) and find F0(1).(b) Let G(x) =g(x)f(x)and find G0(1).(c) Let H(x) = f(x)g(x)h(x) and find H0(1).(d) Let J(x) =f(x)g(x)h(x)and find J0(1).Math 140 Written Homework 6: 3.3, 3.4, 3.5, 3.6, 3.7 Page 2 of 2Question 7 A mass is hanging from the ceiling on a spring and oscillating (forever, and without friction). At itshighest point, the mass is 12 cm from the floor, and at its lowest point, the mass is 6 cm from thefloor. At t = 0, the mass is at its equilibrium and moving downwards, and a single period takes 0.5seconds. HINT: Graph your y(t) to check your answers.(a) Find a formula for y(t), the position of the mass from the floor as a function of time.(b) What is the position, velocity, and acceleration of the object at t = 2?(c) In its first period, during what time interval(s) is the speed, |v|, of the mass less than 1 cm/s?Question 8 The fuel efficiency of a car E(s), in miles per gallon, is a function of the speed of the car s, in milesper hour. Suppose that E(60) = 40 and E0(60) = −0.7.(a) What are the units of E0(s)?(b) Let C(s) be the fuel consumption, in gallons per mile, of the car when it is driving s miles perhour. Find a formula for C(s) in terms of E(s). Calculate C0(60).(c) Let R(s), be the rate the car burns fuel, in gallons per hour, when the car is driving s miles perhour. Find a formula for R(s) in terms of E(s). Calculate R0(60).Question 9 In 1999, the population of Richmond-Petersburg, Virginia, metropolitan area, was 961,400 and wasincreasing at a at roughly 9200 people per year. The average annual income in the area was $30,593per capita, and this average was increasing at about $1400 per year. Use the product rule to estimatethe rate at which total personal income was rising in the area at this time. Explain the meaning ofeach term in the product rule.Question 10 The revenue per month earned by the Couture clothing chain at time t isR(t) = N (t)S(t),where N (t) is the number of stores and S(t) is the average revenue per store per month. Couturereleases a statement to their investors saying that currently they have 50 stores with a total revenueof $7,500,000 and expect to increase the number of stores by about 2 stores per month and increasetotal revenue by about $250,000 per month. What does this statement imply about the current rateof change of the average revenue per store per