Calculus Packet 10 Part 1 Answer Key

A Calculus BC #(1-20) calc $#03-29 31, 32 Recne #30, 38, 46 #45 All problems are NO CALCULATOR unless otherwise indicated. Parametric Curves and Derivatives 1. In the xy-plane, the graph of the parametric equations x = 5t + 2 and y = 3t for - 3 <= t <= 3 isa line segment with slope. A 3/5 B) 5/3 C) 3 D) 5 B2. If x = t ^ 3 - t and y = sqrt(3t + 1) then d/dx (y) att t = 1 is ( 3 * (1) ^ 5 A) 1/8 B 3/8 D) 8/3 E) 13 E) 13 Reuse #34143 OR 3 εστεω α4 98 49 45 3 board Problems 3. A curve in the plane is defined parametrically by the equations x = t ^ 3 + t and y=t^ 4 + 2t ^ 2 .An equation of the line tangent to the curve at t = 1 is: A) y = 2x C) y = 2x - 1 E) y = 8x + 13 B) y = 8x D) y = 4x - 5 3-446 D 4. Consider the curve in the xy-plane represented by x = e ^ 1 and y = t * e ^ (- t) for t >= 0 The slope of the line tangent to the curve at the point where x = 3 is (Calculator) ?=? A) 20.086 B) 0.342 C) -0.005 ( D)-0.011 E) -0.033 A 5. If x = t ^ 2 + 1 and y = t ^ 3 then (d ^ 2 * y)/(d * x ^ 2) = A) 3/(4t) B) 3/(2t) C) 3t D) 6t E) 3/2 12206+1111 1.011 6. A particle moves along the curve xy = 10 1/(x ^ 2) = e_{x}/(x ^ 2) x = 2 and d/dt (y) = 3 what is the value of (dx)/(dt) ? A) - 5/2 B) - 6/5 C) 0 D) 4/5 E) 6/5 E7. If x = e ^ (2x) and y = sin(2t) then A) 4e ^ (2t) * cos(2t) B) (e ^ (2z))/(cos(2t)) C) (sin(2t))/(2e ^ (2t)) D) (cos(2t))/(2e ^ (2t)) (cos(2t))/(e ^ (2t)) of 3/4 d/dx (y) = m rangle= k/a (3/2 * (97 + 1) ^ - 6)/(37 ^ 2 - 1) d/dt (x) = x_{y}/x_{y} cos 2k 2e ^ (3t) (dy)/(dx) = nabla t 2 = nabla c 2 k = 2/(4t) y' = - y' m = 0 d/dx (y) = - 7/(e ^ (2x)) + 1/(e ^ - 7) y = 0 d/dx (u) = (4(- c ^ - 2) + c)/(e ^ x) (dy)/(dx) = (17 ^ 9 + 4z)/(2t ^ 9 + 1) = 118. 4 C 8. For what values of t does the curve given by the parametric equations x = t ^ 3 - t ^ 2 - 1 and y = t ^ 4 + 2t ^ 2 - 8t have a vertical tangent? (dx)/(dt) = 37^ 2 -27=0 7(s)-2)= ?= 0 A) 0 B) 1 t(Bk - 2) = 0 D) 2/3 and 1 E) No value D9. For 0 <= t <= 13 an object travels along an elliptical path given by the parametric equations x=3cost and y = 4sin t At the point where t = 13 the object leaves the path and travels along the line tangent to the path at that point. What is the slope of the line on which the object travels? A) - 4/3 B) - 3/4 C) - (4tan 13)/3 4 3 tan 13 D) E) - 3/(4tan 13) C 10. The position of a particle moving in the xy-plane is given by the parametric equations x = t ^ 3 - 3t ^ 2 y = 2t ^ 3 - 3t ^ 2 - 12t For what values of t is the particle at rest? A)-1 B) 0 C) D)-1,2 A (\mathfrak{G}*y)/x = 6T - \langleg\rangle - 16 rangle E)-1, 0, 2 11. A curve C is defined by the parametric equations x = t ^ 2 - 4t + 1 and tilde y = t ^ 2 . Which of the following is an equation of the line tangent to the graph of C at the point (-3, 8)? t = y ^ (1/2) 12 Vertical som 4-4 A) x = - 3 B) x = 2 NOT - 3 = t ^ 22 - 4(t + 1) 7-47+4 C) y = 8 D) y = - 27/10 * (x + 3) + 8 E) y = 12(x + 3) + 8 12. (1973 BC4, p. 538 #33) A kite flies according to the parametric equations x = t/8 and y = - 3/64 * t(t - 128); u = - 2 where t is measured in seconds and 0 < t <= 90 cket acket (a) (b) At what rate is the kite rising at t = 32 seconds? 3 How high is the kite above the ground at time t = 32 seconds? 144 (c) At what rate is the string being reeled out at t = 32 seconds? 3,002 y (d) At what time does the kite start to lose altitude? z = 64 X 13. (1974 BC5) Given the parametric equations x = 2(t - sin t) and y = 2(1 - cos t) (a) Find d/dx (y) in terms of t (b) Find an equation of the line tangent to the graph at t = pi y = 4 (c) Find an equation of the line tangent to the graph at/ = 2pi . x = 471 (d) d/dt (y) = - 3/32 * z + 6 d/dt (x) = 1/8 y = - - 3/64 * t ^ 2 + 6t dx = a * x ^ 2 - ax 0 = 9/1 - 2 2/3 o = 6(7 ^ 3 - 7 - 2) (sin theta)/(1 - cos theta) d/dx (y) = (4cos 7)/(- 3s * ln(7)) = - 4/3 * cot 7 = - 4/(3 * 7on * 1s) 47. (2006B BC2) An object moving along a curve in the xy-plane is at position ( x(t) , y(t) ) at time t, where d/dt (x) = tan(e ^ (- t)) and d/dt (y) = sec(e ^ (- t)) for t >= 0 At time t = 1 the object is at the point (2,-3). (a) Write an equation for the line tangent to the curve at position (2, -3). (b) Find the acceleration vector and the speed of the object at time t = 1 (c) Find the total distance traveled by the object over the time interval 1 <= t <= 2 (d) Is there a time t >= 0 at which the object is on the y-axis? Explain why or why not. 48. (2007B BC2) An object moving along a curve in the xy - plane is at position (x(t), y(t)) at time t with (-3,4). and for t >= 0 At time t = 0 the object is at position d/dt (x) = arctan(t/(1 + t)) * i d/dt (y) = ln((t ^ 2 + 1)!) (a) Find the speed of the object at time t = 4 (b) Find the total distance traveled by the object over the time interval 0 <= t <= 4 (c) Find x(4) (d) For t > 0 there is a point on the curve where the line tangent to the curve has slope 2. At what time t is the object at this point? Find the acceleration vector at this point. 49. (2008B BC1) A particle moving along a curve in the xy - plane has position ( x(t) , y(t) ) at time t >= 0 with and The particle is at position (1, 5) at time t = 4 d/dt (y) = 3cos((t ^ 2)/2) d/dt (x) = sqrt(3t) (a) Find the acceleration vector at time t = 4 (b) Find the y coordinate of the position of the particle at time t = 0 (c) On the interval 0 <= t <= 4 at what time does the speed of the particle first reach 3.5? (d) Find the total distance traveled by the particle over the time interval 0 <= t <= 4 Answers Parametric Curves and Derivatives Length of Parametric Curves 1. A 1998 BC #2 92% 18. D 1993 BC #23 83% 2. B 1985 BC #30 73% 19. D 1997 BC #15 66% 3. C 1988 BC #34 68% 20. C 1998 BC #21 73% 4. D 1993 BC #25 54% 5. A 1993 BC #6 39% Vectors 6. B 1993 BC #4 75% 23. E 1998 BC #77 87% 7. E 1997 BC #2 92% 24. D 1985 BC #4 69% 8. C 1997 BC #18 62% 25. E 1988 BC #15 75% 9. D 2003 BC #4 79% 26. A 1993 BC #28 86% 10. C 2003 BC #7 29% 27. E 1998 BC #10 75% 11. A 2003 BC #17 36% 28. C 2003 BC #84 62%