Mathematics 1A Spring 2010 A Ogus Sample Midterm Exam 2 Instructions Closed book exam No formula sheets or notes are permitted Calculators and other electronic devices are not allowed Turn cell phones off and stow them in backpacks pockets purses Show work and or reasoning where indicated 1 Calculate f 0 x using any method from this course Show your steps 1a f x x 1 e2x 1b f x p xln x 1c f x arcsin x 1d If x4 xy 2y 4 20 find dy dx when x y 2 1 2a Find the maximum value of f x x x 1 2 on 1 2 and determine all points 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 possible value of x 2y Show your steps 3 Suppose that f and f 0 are differentiable functions on an interval a b c a b and f 0 c 0 What can one conclude if f 00 c 0 What if f 00 c 0 Answer both questions 4 Let f x x1 3 What is the equation for the linearization also known as the linear approximation of f at 8 5 Show that if x 1 then ln x x 1 1 2 Sample problems and study guide Calculation problems Calculate f 0 x f x x3 8x2 x93 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 final exam Exponential growth decay A certain radioactive substance has a half life of 1 000 years If a chunk of this substance weighs 1 gram today how much radioactive material will remain after 2 500 years It is not necessary to simplify your answer in any way Curve sketching This is a very important topic but because these problems take a significant amount of time you are not likely to be asked to draw complicated graphs on the midterm exam You may be asked other types of questions designed to test individual steps in the curve sketching process Related rates problems See the examples treated in lectures and those assigned for 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 different sections of our text Optimization problems are word problems which involve additional steps using constraint equations to eliminate independent variables in order to obtain a maximization or minimization problem Solution of optimization problems sometimes requires additional skills including curve sketching and analysis of 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 f 0 Show that sin x x for all x 0 Suppose that f 0 x x 2 x 1 2 x 3 x 5 On which intervals is f an increasing function d Assuming that ln is a differentiable function show that dx ln x 1 x Theory questions State Rolle s Theorem If f has a local minimum at c and if f 0 c exists show that f 0 c 0 3 If f 0 x 0 for all x in an interval then f is increasing on that interval How do we know this fact If f is a differentiable function on an interval a b and if f 0 x 0 for every x 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 T 0 t0 0 and T 00 t0 0 At a slightly later time t1 t0 how would we expect T t1 and T 0 t1 to be related to T t0 and T 0 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 x near a Be prepared to give bounds for the error your may be making and to predict if 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 time t 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 maximum possible number of times t 0 at which r t h t Explain your answer briefly 4a Define f has an inflection point at x 1 5 sec x cos x with domain 0 2 in this problem Let arcsec be the inverse function of sec Assuming that arcsec is differentiable show that arcsec0 x x x12 1 for x 0 Show your reasoning
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