MAC2311 - Note Problems 1—8 are worth four points each 2...

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Unformatted text preview: Note: Problems 1—8 are worth four points each. 2 . 1. Suppose we know that 6 is an angle in Quadrant III and that tan(6) = 5 . Find the value of sin(26). 12 6 4 6 12 A. — B —— C ——-— D. —— - 13 «1‘3 m 13 5 . . . . 2a: — x2 ( 2. Wthl'l of the followmg statements 13/ are true of the function f(:1:) = m ? I. f(:23) has no x-intercepts. II. f(a:) has no vertical asymptotes. III. f (ac) has no removable discontinuities. A. II only E. III only C. I and II only D. I and III only E. I, II, and III 3. The graph of which of the following functions is the graph of f (3:) shifted down two units, then shifted left two units, then reflected over the :c-axis, then reflected met the y—axis‘? A. y=—2‘f(1'+2) B. y=~2+f(z-2) C. y=2—f(—2—:r) D. y=2—f(2—:z) E. y=—2—f(2—:n) 4. The area and perimeter of a certain rectangle have the same numerical value. Express its length 2 as a function of its width 4». E. This cannot be done. 5. A current population of 17 mice will double every 10 days. Express the population of mice t days from now as a function m(t) . AA ma) = 17(2)r‘a B. m(t) = 10(2)”‘ 0. m(t) = 10(17)2= D. me) = 17(10)2t E. m(t) = 17(10)é 6. Which of the following statements is/ are true of the function below? 2 x<~1 1 f(a:)= 1*~ —1<a:51 a; 1+fi a:>1 L f (3:) has one jump discontinuity. II. f (3:) has one infinite disoontinuity. III. f (at) has one removable diacontinujty. A. I only B. III only C. I and II only D. II and III only E. I, II, and III 7. Evaluate the limit: lim 1 l 1 , _ . —- D. — E. 00 A. 0 B 4 C fl 2 1 . . 8. Examine the functions f(a:) = E and g(::;) = 1 + fl . Which one of the functions listed below DOES NOT have domain (0, oo ) ‘? I.fg 11% III.gof IV.fog V.f+g A. I B. II C. III D. IV E. V Note: Problems 9—12 are worth two points each. 9. Which of the following identities are true for all positive values of 3:? 1. ln( ex + 2) = a: + 111(2) 11. (111(1))3 = 31n(a:) 111. (g): = 3-1 A. II only B. III only C. I and II only D. I and III only E. I, II, and III 60 10. The function g(a:) = —— 10+ on [0.25] is one—to-one. Evaluate 574(8). 1+\/5 A. 1 B. 4 C. 9 D. 16 E. 25 11. Choose the function that is NOT oue—to—one: A. Mm) = 6"” B. h(z) *4: <1«o=x3 rxflo=v6 EA all are these functions are one-to—one 12. Determine the value of the constant c for which the following piecewise—defined func- tion is continuous: lvfi x>l flfl— l'x 1+0 a:<l 1 1 A~2 B—— .— -2 2 00 D2 E Bonus!l(4 points) 13. Consider the functions below that we analyzed in lecture. How many of the functions have a vertical asymptote at z = O 7 sin(a:) __ [m] :5 III. h(2:) w 7 I. m) = :52 cos(%) II. 57(3)) = A.1 B2 0.3 D.0 MAC 2311 —-- EXAM 1 Free Response NAME SECTION UF lD TEST FORM CODE A YOU MUST SHOW ALL OF YOUR WORK TO RECEIVE C-REDITLI 1. Suppose a particle travels in A straight line so that. its position in meters after t seconds is given by 5(2‘) 2 t — £2 . a) Calculate thc average velocity of the particle during the first 2 seconds of it: motion. Then calculate the average velocity on the interval [2,3]. Include units with your answer b) Using limits, find the instantaneous velocity of the particle when exactly '2 seconds have passed. Include units with your answer. 2. C irclc all intervals below on which Intermediate Value Theorem guarantees a 2 I v 5 root. (zero) of the function = 1 . — :1: I. (—4, #2) 11. (—2.0) 111.012) IV. (2,4) 3. SOLVE TWO of the. following equations /inequalities for 1?. Write the solutions as a set in the manner indicated below You may solve all three if you have time for bonus points. a) W’rite the solution set: log4(.r - 2) = log4(2 + l) + 2 b) Write the set of solutions in [0, 27v ): tan(ar) : 25in(.r) (3) Write the solution set in interval notation: NAME SECTION 4. Examine the function f (x) 2 13h}; f(I-‘) =_ 33;): f (I) =__ 11331 f (I) =_ lim f(r) =__ lim f(x) =___ lim =__ rm—Q' .r—«>0+ $—>2+ 11392 M) :__ gig M) =r gig f(:r) z- b) Sketch the graph of fix) on the axes below. (Consider simplifying the expres— sion first) c) List all values of a: at. which f (:13) is discontinuous, and classify each as a remov- able, jump. or infinite discontinuity. 5. A population of wolves increases so that the number of wolves x years from now is _:r, given by the funcrion W(:r) : a( l — e E j" . 21) Write the formula for l’V(I) when a = 2, b = 4, and n = 3. Then calculate its inverse function W‘l(.r) . b) Write the formula for W(;r) when a = b = n = 1. Then calculate its inverse function W ‘1 (:r) i c) Sketch the graphs of W(.r) and its inverse on the same axes below for the case when u = l) = -n. = 1 as in part (b) For the sketch, you may assume the domain of W(:r) is ALL REAL NUMBERS. Label any usymptotes and intercepts ...
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