1-Variable Calculus Review for Math 208• Work all problems on scratch paper until you are sure they are correct. Then copy yourwork onto these pages. If you need a clean copy, you can find one athttp://www.math.unl.edu/˜gledder/Math208/calcrev.pdf.• Make sure you show sufficient work and give sufficient explanations where needed.• These problems are representative of the skills and concepts from 1-variable calculus thatare needed in Math 208. You are encouraged to discuss the problems with classmatesand to look things up in your calculus book. You are welcome to ask your instructor forhints. DO NOT COPY THE WORK OF OTHERS.1. (6 points each) Find and simplify the derivatives of the following functions (you must showyour work):(a) f (x) = 4x2ln(cos x)(b) f (x) =qsin(e−3x)(c) f (x) =x24x + 112. Compute the following integrals or explain why they do not exist (you must show yourwork):(a) (8 points)Z21x2x3+ 1dx(b) (6 points)Zx3+ 1x2dx(c) (6 points)Zsin3x cos x dx(d) (6 points)Z(4 − 3x)e2xdx23. (6 points each) The graph shown below is that of the derivative of a function g. Use it toanswer the following questions about g.Graph of g0-3-2-101234012 34 5678 9 10(a) Compute g(9) − g(6).(b) At what values of x does g have local maxima? local minima? Explain your answer (youmust indicate which is which and how you know).(c) Suppose g(5) = 2. Use a tangent line approximation to estimate g(0).(d) Sketch the graph of g00below.-3-2-101234012 34 5678 9 1034. (8 points each) (a) Find the average value of the function exon the interval 0 < x < 1.(b) Suppose I currently owe you $100 and the debt increases at a continuous rate of 10t dollarsper month, where t is the time in months. How much will I owe you after 1 month?5. (8 points) Let a be a positive number and let f (x) = x3− ax2+ 1. Find the critical pointsof f and classify each as a maximum, a minimum, or an inflection point.6. (8 points) Suppose f0is defined for all values of x, is 0 for x = 3 and x = 4, and is not0 otherwise. Suppose further that it costs $1 to measure a value of f for any x. How muchdo you have to spend to find the maximum value achieved by f on the interval 0 ≤ x ≤