Mathematics 1A, Spring 2010 — A. OgusSample Midterm Exam #2Instructions.Closed book exam — No formula sheets or notes are permitted. Calculators andother electronic devices are not allowed. Turn cell phones off and stow them inbackpacks/pockets/purses.Show work and/or reasoning where indicated.(1) Calculate f0(x), using any method from this course. Show your steps.(1a) f(x) = x−1e2x(1b) f(x) = xln(x)(1c) f(x) =parcsin(x).(1d) If x4+ xy + 2y4= 20, find dy/dx when (x, y) = (2, 1).(2a) Find the maximum value of f(x) = x(x − 1)2on [−1, 2], and determine allpoints in this interval where that value is attained.Show all steps; you will be graded on these steps, not merely on your answer.(2b) If x and y are positive numbers and xy = 1, what is the minimum possiblevalue of x + 2y?Show your steps.(3) Suppose that f and f0are differentiable functions on an interval (a, b), c ∈ (a, b),and f0(c) = 0. What can one conclude if f00(c) > 0? What if f00(c) = 0? (Answerboth questions.)(4) Let f(x) = x1/3. What is the equation for the linearization (also known as thelinear approximation) of f at 8?(5) Show that if x > 1, then ln(x) < x − 1.12Sample problems and study guideCalculation problems:( ) Calculate f0(x):( ) f(x) = x3− 8x2+9x3.( ) f(x) = arctan(x2+ 1)( ) f(x) = ln(|x|)Application problems:In general, this exam will not stress applications at all. (Because of time constraints;it takes longer to read and set up word problems. Expect to see these on the finalexam.)( ) Exponential growth/decay. A certain radioactive substance has a half life of1, 000 years. If a chunk of this substance weighs 1 gram today, how much radioactivematerial will remain after 2, 500 years? (It is not necessary to simplify your answerin any way.)( ) Curve sketching. This is a very important topic, but because these problemstake a significant amount of time, you are not likely to be asked to draw complicatedgraphs on the midterm exam. You may be asked other types of questions, designedto test individual steps in the curve sketching process.( ) Related rates problems. See the examples treated in lectures, and those assignedfor homework. Many additional examples can be found in our text.( ) Max/min problems, and optimization problems. See examples treated in lecture,and assigned problems. See text for additional problems. (These were two differentsections of our text. Optimization problems are “word problems”, which involveadditional steps: using constraint equations to eliminate independent variables, inorder to obtain a maximization or minimization problem. Solution of optimizationproblems sometimes requires additional skills, including curve sketching, and analysisof asymptotic behavior.)Calculation/theory problems:( ) Suppose that f is a differentiable function on the interval [0, 2], that f(0) = 0,and that f(1) = −6. What can we conclude about the range of the function f0?( ) Show that sin(x) < x for all x > 0.( ) Suppose that f0(x) = (x + 2)(x − 1)2(x − 3)(x − 5). On which intervals is f anincreasing function?( ) Assuming that ln is a differentiable function, show thatddxln(x) = 1/x.Theory questions:( ) State Rolle’s Theorem.( ) If f has a local minimum at c, and if f0(c) exists, show that f0(c) = 0.3( ) If f0(x) > 0 for all x in an interval, then f is increasing on that interval. How dowe know this fact?( ) If f is a differentiable function on an interval (a, b), and if f0(x) = 0 for everyx ∈ (a, b), then f is constant. How do we know this?( ) Let t = time, and let T (t) = the temperature of a liquid at time t. At time t0,measurement reveals that T (t0) = 75, T0(t0) < 0, and T00(t0) > 0. At a slightly latertime t1> t0, how would we expect T (t1) and T0(t1) to be related to T (t0) and T0(t0),respectively?( ) Define: The graph of f is concave up, over the interval (a, b).( ) Use the linearlization of a differentiable function f at a to estimate f(x) for xnear a. Be prepared to give bounds for the error your may be making, and to predictif the estimate is too small or too large.(4e) A rabbit and a hare race along a straight line, beginning at time t = 0. At timet, their positions are r(t) and h(t), respectively. Suppose that r(t) = 1 + t for all t(the rabbit is given a head start), while h00(t) < 0 for all t. What is the maximumpossible number of times t > 0 at which r(t) = h(t)? Explain your answer briefly.(4a) Define: f has an inflection point at x.(5) sec(x) =1cos(x), with domain (0, π/2) in this problem. Let arcsec be the inversefunction of sec. Assuming that arcsec is differentiable, show that arcsec0(x) =1x√x2−1for x > 0. Show your