Math Calculus I - SECTIONS A and B SAMPLE FINAL EXAMINATION Thursday, May 3rd, 0 00 POSSIBLE POINTS DISCLAIMER: This sample eam is a study tool designed to assist you in preparing for the final eamination in Harrison s sections A and B of Math. Although these problems are representative of the type of questions you may be asked on the final eamination, they do not, in any way, represent an ehaustive list of the possible questions that may appear on that test. I. (0 points) OBJECTIVE QUESTIONS () If f and g are differentiable on the real line, f() = ; f() = ; g() = ; g() = ; f' () ; f' () 5; g' () 4; and g' () 6, then (g(f()))' (a) (b) 4 (c) (d) 8 (e) 30 (f) none of the above () Which of the following functions are anti-derivatives of f() sin 3 ()cos() on the interval [,00]? (a) 4 sin 4 (b) - 3 sin (t)cos(t)dt e (c) each of (a) and (b) (d) neither (a) nor (b) (3) (TRUE OR FALSE) If f is differentiable at a real number a, then f is continuous at a real number a. (4)(TRUE OR FALSE) L Hopital s Rule can be applied to show that if f() = cos()/, then f ' () - sin() - sin().
II. (4 points) Compute the following limits.. lim 4 4-0. Tan lim - () cos() -3 3. lim e 4. lim (8 7) 0
3 III. (6 points) Below are two graphs. One is the derivative of the other. Identify which is which. IV. (6 points) Consider the graph of the following function f below. GRAPH OF f a b (a) Does lim f() a eist? Is f continuous at a? Does lim f() b eist? Is f continuous at b? (b) Does lim f() a - f(a)? Does lim a f() f(a)? Does lim b - f() f(b)? Does lim b f() f(b)?
4 V. (30 points) In parts () (5), find the derivative of the specified function/relation with respect to (SHOW YOUR WORK JUSTIFY YOUR ANSWERS). (a) y ( - )( ) (b) y 3 sin(e ) (c) F() 3 5 ( -) (e ) ln( ) 5 4 ( ) ( ) (d) 5 3 y 4 (e) y () 3
5 VI. (3 points) () (8 points) Use the DEFINITION of DERIVATIVE to compute the derivative of the function f() = /( 3) for any in its domain. () (5 points) Utilize your work in part () to find the equation of the tangent line to f at the point P = (,-).
6 VII. ( points) Consider the function f() = (3/) on the interval [0,]. (a) Approimate the integral of f over the interval using one right-hand rectangle (in this case the rectangle is inscribed). (b) Approimate the integral of f over the interval using 00 righthand/inscribed rectangles (HINT: Setup an appropriate Riemann sum and enter it into your TI-89 graphing calculator. The technique you utilize for part (b) should allow you to complete part (a)). (c) Apply the Fundamental Theorem Of Calculus s Evaluation Theorem directly to calculate
7 VIII. (4 points) Calculate the following definite/indefinite integrals eactly (YOU MUST SHOW ALL WORK INCLUDING ALL SUBSTITUTIONS. DO NOT APPROXIMATE USING YOUR CALCULATOR). () ((5.) 5. sec ())d () 0 e 3 d (3) 6 t dt t 3 (4) sin 3 () d
8 IX. (0 points) Consider the following function. f() = - () Find the domain of f. Show that f is positive on its domain. () Find the horizontal asymptote(s) of f. (3) Determine any critical points of f. Determine any local maimum or minimum values of f. Find intervals where (i) f is increasing, (ii) f is decreasing. (4) Determine any inflection points of f. Find intervals where (i) f is concave upward, (ii) f is concave downward. (5) Use your previous answers to sketch f.
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0 X. (35 points) Application Problems -- complete eactly FOUR of the FIVE problems -- SHOW YOUR WORK to OBTAIN FULL CREDIT Fifth Problem is a bonus problem for up to NINE POINTS. () An inverted conical water tank with a height of feet and a radius of 6 feet drains through a hole in the verte at a rate of ft 3 /s. What is the rate of change of the height of the water in the tank when the water is 3 feet deep? NOTE: The volume of a cone is given by V = /3( r h) where V is the volume of the cone, r is the radius of the cone, and h is the height of the cone. () The aerobic rating of a person years old is modeled by the function - for > 0. At what age is a person s aerobic rating the largest? (3) Assume a stone is thrown vertically upward with an initial velocity of 64 ft/s from a bridge 96 feet above a river. What is the maimum point above the river reached by the stone? With what velocity will the stone strike the river? (4) An oral painkiller is administered to a patient. After t hours, the concentration of drug in the patient s bloodstream is given by C(t) = t/(3t + 6) mg. At what rate is the concentration of drug changing after the first hour (in mg/hr)? Is the drug concentration changing at an increasing rate or a decreasing rate at this time (EXPLAIN)? (5) Assume the length L(t) of a certain organism at time t (t is in years) satisfies the following differential equation dl/dt = e -0.t where t > 0 Assume that (a) Find the function L(t). (b) Compute L(0) (the initial length of the organism).
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