Final Exam. Math 3 Daugherty March 13, 2012

Transcription

1 Final Exam Math 3 Daugherty March 3, 22 Name (Print): Last First On this, the Final Math 3 exams in Winter 22, I will work individually, neither giving nor receiving help, guided by the Dartmouth Academic Honor Principle. Signature: Instructions: You are not allowed to use calculators, books, or notes of any kind. All of your answers must be marked on the Scantron form provided. Take a moment now to print your name and section clearly on your Scantron form and on the cover of your exam booklet and sign the a rmation. You may write on the exam, but you will only receive credit on Scantron (multiple-choice) problems for what you write on the Scantron form. At the end of the exam, you must turn in both your Scantron form and your exam booklet. There are 25 multiple choice problems worth each, for a total of 25.

2 . If f(x) = x 4, whatcanyousayabout lim h! f(4 + h) f(4)? h f(4 + h) (a) It doesn t exist because the denominator is going to, so f(4) h f(4 + h) f(4) f(4 + h) f(4) (b) It doesn t exist because lim 6= lim. h! + h h! h (c) It s equal to ±. (d) It s equal to. goes to infinity. 2. Which of the following is true if f(x) = ( x 3 + x + x> cos(x) x apple? (a) This function is continuous and di erentiable everywhere. (b) This function is continuous everywhere, and di erentiable everywhere except at x =. (c) This function is continuous everywhere except at x =,butisdi erentiableeverywhere. (d) This function is continuous and di erentiable everywhere except at x =, where it is neither. ++ = cos()= continuous! 3()^2 + = - sin() = not differentiable! 2

3 (x+2)(x- 2) 3. Let f(x) = x2 4. Which of the following is true? x 2 2x x(x- 2) (a) This function has one horizontal asymptote and one vertical asymptote. (b) This function has one horizontal asymptote and two vertical asymptotes. (c) This function has no horizontal asymptotes and one vertical asymptote. (d) This function has no horizontal asymptotes and two vertical asymptotes. H: x=, V: y= 4. Use the linear approximation of ln(x) ata =toestimateln(5). (a) 2 (b) 4 (c) 5 (d) + (4) 5 y = /x L(x) = /(x- ) + ln() = x- L(5) = 4 3

4 5. Which of the following describes the graph of f(x) =x 3 +6x 2 +9x? (a) It is positive exactly over (, ), increasing exactly over (, 3), and concave up exactly over ( 2, ). (b) It is positive exactly over (, ), increasing exactly over (, 3) [ (, ), and concave up exactly over ( 2, ). (c) It is positive exactly over (, 3) [ (, ), increasing exactly over (, 3) [ (, ), and concave up exactly over ( 2, ). (d) It is positive exactly over (, 3) [ (, ), increasing exactly over (, 3) [ (, ), and concave up exactly over ( 2, ). f = x(x+3)^2 > when x> f = 3x^2 + 2 x + 9 = 3(x+3)(x+) > when x>- and x<- 3 f = 6x+2 = 6(x + 2) > when x>- 2 4

5 6. First ask yourself how many points on the curve y = - 4x^3/3y^2, so V: x=, H:y= H: If x =, y^3 = -, so y=- (one solution) V: If y=, x^4=-, so no solution. y 3 + x 4 = have vertical or horizontal tangent lines, respectively. Now, based on that information, which of the following could be a graph of this curve (plotted on the usual x-y axes, but shown without those axes)? (a) (b) (c) (d) (e) None of the above. 5

6 7. What is the maximal value of ( + x 2 ) /2 on [, 2]? (a) (b) 2 (c) p 5 (d) p 5 f = x/sqrt( + x^2), so the only c.p. is at x= f() = f(- ) = sqrt(2) f(2) = sqrt(5) MAX! 8. Suppose a parked car is leaking oil in such a way that the oil is leaving a very thin circular puddle. If the area of the puddle is growing at a rate of 3 cm 2 /hr, how fast is the puddle s radius growing when the puddle is 2cm wide (r =cm)? (a) 2 (b) 3 2 (c) 3 5 cm/hr (d) 6 cm/hr A = 3 want r () A = pi*r^2 so A = 2pi*r * r so when r=, 3= 2pi**r, and so r = 3/2pi 6

7 9. Suppose a ball is tossed upwards at a velocity of 7m/s from a height of 2 meters above the ground. Neglecting wind/air resistance, what is happening to the ball exactly second later? (The acceleration due to gravity has a magnitude of 9.8 m/s) (a) It is still moving upwards. (b) It is falling, but is still above its initial position. (c) It is falling, but is below its initial position. (d) It has already hit the ground, so we don t know for sure. y()=2, y ()=7 y = - 9.8t + 7, y=- 4.9t^2 + 7t + 2 y () <, y() >2 7

8 . If f(x) =5 x ln(3x), what is f (x)? (a) f (x) =5 x ln(3x) (b) f (x) =5 x x (c) f (x) = ln(5)5 x (d) f (x) = 5x ln(5) 3x x. If f(x) =cos(x)e sec(x)+2,whatisf (x)? (be mindful of small simplifications) (a) f (x) = (b) f (x) =(cos(x) sin(x)e sec(x)tan(x) sin(x))e sec(x)+2 (c) f (x) = cos(x) (sec(x)tan(x)+2) (d) f (x) =(tan(x) sin(x))e sec(x)+2 sin(x) e sec(x)+2 (- sin(x) + cos(x)sec(x)tan(x))exp(sec(x)+2) 8

9 2. Which of the following is equal to (a) 4X arctan(i/4) i= (b) lim n! (c) lim n! (d) lim n! nx i= nx i= nx i= n arctan(i/n) 4 n arctan(4i/n) 4 n arctan(4i/n) no limit Z 4 arctan(x) dx? 3. Approximate I = R 8 x cos(x) dx using three equally spaced rectangles and left-sided endpoints. 2 (a) I 2cos(2)+4cos(4)+6cos(6) (b) I 4cos(4)+6cos(6)+8cos(8) (c) I 4cos(2)+8cos(4)+2cos(6) (d) I 4cos(2)+8cos(4)+2cos(6)+6cos(8) Dx = 2, evaluate at 2,4,6 9

10 4. Assume a population of lemmings is growing at a rate proportional to the number of mice present. If there are 5 lemmings now and lemmings in 6 months from now, now many lemmings to you expect there to be in a year? (a) 5 5 (b) 5 5 (c) 5 e (d) y = Ae^(kt) y() = 5 y(6) = (error) 5. Solve dy dx = y3 - /(2y^2) = x+ c (a) x = 4 y4 + C. (b) y = p 2(x + C) (c) y = p + C 2x (d) y = 3p e x+c

11 6. Which of these di erential equations has the following slope field? (a) dy dx = x + y (b) dy dx = x (c) dy dx = xy (d) dy dx = x y y 7. Use Euler s method with step size x =2toapproximatey(4) if dy dx = y x and y() = 5. (a) y(4) 2 (b) y(4) 4 (c) y(4) 43 (d) y(4) 5 x =, y=5, m = 5, y= 5+ 2*5 = 5 x = 2, y = 5, m= 5-2=3, y2= 5 + 2*3 = 4.

12 8. If f (x) = sin(x)+x + e 5x,whichofthefollowingcouldbef(x)? (a) f(x) =cos(x) x 2 +5e 5x (b) f(x) =cos(x)+ln x +5e 5x (c) f(x) = cos(x)+ln x + 5 e5x (d) f(x) = cos(x) ln x + 5 e5x 9. Which of the following is equal to the arclength of p x from x =3tox =5? Z 5 r (a) 3 4x dx Z s 5 (b) + 2 p x dx (c) (d) 3 Z 5 3 Z p x dx p xdx 2

13 Z e x 2. Calculate p dx +e x (a) p +e p 2 (b) 2 p +e 2 p 2 (c) 2e p +e 2 p 2 e (d) p +e u = + exp(x), du = exp(x) dx int u^(- /2) du = 2 sqrt(u) + C = 2 sqrt( + exp(x)) + C 3

14 2. What are the (standard) domain/range of y =arccot(x)? Text (Hint: Remember arccot(x) = cos(x)/ sin(x) has a vertical asymptote every time sin(x) =.) (a) Domain: all <x<,range:ally. (b) Domain: all x, Range:<y<. (c) Domain: all x, Range: /2 <y<. (d) Domain: all x, Range: ally. 22. Which of the following integrals would you use to calculate the area under y =arcsin(x)between x =andx =? Your choice should be one which you would be able to compute exactly using the techniques from class, without having to rely on approximations. (a) (b) (c) (d) Z Z /2 Z Z /2 arcsin(x) dx sin(y) dy sin(y) dy sin(y) dy 4

15 23. Which of the following is equal to the area between f(x) =x 3 3x and g(x) =x? (a) (b) (c) (d) Z 2 2 Z 2 Z (x 3 4x) dx (x 3 4x) dx (x 3 4x) dx + 2 Z (x 3 p 3 4x) dx Z 2 Z 2 (x 3 4x) dx (x 3 4x) dx Z p 3 (x 3 4x) dx f- g solve f=g, so =f- g = x^3-4x = x(x- 2)(x+2) f g g f 5

16 For the next two problems, suppose f(x) andg(x) arethefunctionspicturedbelow: odd f(x) even g(x) Which of the following is not an even function? (over the domain shown) (a) g(3x) (b) f(g(x)) (c) g(x)+. (d) More than one of the above is not even. (e) All of (a)-(c) are even. 25. Notice the f(x) g(x) is an odd function. What can you say about the integral Z f(x) g(x) dx? (a) It s equal to. (b) We don t have enough information to calculate it exactly, but it is equal to 2 Z f(x) g(x) dx. (c) We don t have enough information to calculate it exactly, but it is equal to Z f(x) dx Z g(x) dx. (d) We don t have enough information to calculate it exactly, but we know it s positive. (e) There s simply not enough information to make any conclusion. 6

17 (For scratch work) 7