MULTIVARIABLE CALCULUS

Transcription

1 MULTIVARIABLE CALCULUS Summer Assignment Welcome to Multivariable Calculus, Multivariable Calculus is a course commonly taken by second and third year college students. The general concept is to take the foundations of one variable calculus (AP Calc), and extend them to include functions of several variables. Due to this relationship, a MASTERY of AP Calculus concepts is absolutely imperative in order to be successful. Therefore, the purpose of this assignment is to have you practice the skills necessary to be successful in Multivariable Calculus. All of the skills in this packet are skills that you should have mastered prior to taking this course. Each question was carefully selected and every question is equally important. You should not use calculators to complete this packet unless it is otherwise specified. Graphing calculators will not be used on any of the tests in this course, and everything you do in class will be completed by hand. This assignment will be collected on the first day, and any portion that I choose, up to and including the entire packet, will be graded. No partial credit will be given. No late submissions will be accepted. You must show ALL WORK to support your answers, and you may attach separate sheets to this packet to do so. However, your final answers must be written next to the questions in this packet. All work shown should be NEAT and ORGANIZED. If I cannot follow it, you will not receive credit. In addition, there may be a diagnostic assessment on this material at any moment during the first week of school. Thank you for your interest in this course, and I will see you in class in September. ~ Mr. Nemeth * If you need to contact me at any point, please send me an at lnemeth@cboek12.org

2 LIMITS Compute the following limits: 6+4x 1) lim = x 3 x 2 2) lim +1 x 5 x 2 25 x 2 +2x 15 = 3) lim 5x2 16x+3 x = x 3 9 x 2 4) lim = x 0 3 x+9 5) lim ( 9 x 3 (x 3) 5) = 6) lim ( 2t ) = t 6 6+t 7) lim x 8x + 9x3 11x 5 = 5x ( 8 9 x x 3 +10x 5 3x 8) lim 8) =

3 9) lim x ( 12+x 3x 2 8x+23 ) = 10) lim ( 25x+7 ) = x 5x ) lim x 0 x sin(7x) cos(2z) 1 = 12) lim = z 0 z 13) lim x 0 3x cos x = 14) lim x 0 2x 2 1 cos = 2 x sin(x 4) 6x 15) lim = 16) lim = x 4 x 4 x 0 tan x

4 CONTINUITY Determine if the functions are continuous at the given point(s): 1) f(x) = { x + 7 if x < 2 5x + 7 if x < 3 7x + 1 if x > 3, at x = 3 2) f(x) = { 9 if x = 2 3x + 3 if x > 2, at x = 2 Determine the TYPE of discontinuity in the following functions (Point, Jump, Asymptotic, or Removable): 3) f(x) = { x2 if x 3 12 if x = 3 4) f(x) = { x + 2 if x 4 x 2 8 if x > 4 Where does the following function have a removable discontinuity? 5) f(x) = 2x2 7x 15 x 2 x 20 For what value(s) of k is the following function continuous? 6x 12 if x < 3 6) f(x) = { k 2 5k if x = 3, at x = 3 6 if x > 3

5 DERIVATIVES Compute the following derivatives by any method: 1) f(x) = 6x 3 9x + 4 2) f(x) = x(3x 2 4) 3) f(y) = y 4 9y 3 + 8y ) f(x) = x + 8 x 4 2 x 5) f(t) = t 6t 3 6) f(z) = t5 z 3 8z 4 3z 10 7) f(x) = (4x 2 x)(x 3 8x ) 8) f(t) = (1 + t )(t 3 ) 9) f(x) = 3x + x4 2x ) f(x) = x +2x 7x 4x 2

6 11) f(x) = cos x sin x, at x = π 12) f(x) = tan 2 x + 9 csc x 13) f(y) = 2e y 8 y, at y = 0 14) f(x) = 3 x log x 15) f(t) = t 5 e t ln t 16) f(z) = 1+5z ln z 17) f(x) = tan 1 (3x + 1) 18) f(x) = 2 1 6x, at x = 0 19) (x+y) (x y) = 3, at (2,1) 20) 7y2 + sin(3x) = 12 y 4, at (0,1) 21) If f(x) = x 3 and g(x) = f 1 (x), find g (8)

7 22) If f(x) = x 3 x 2 + x 1, find f (4) (x) 23) If f(x) = sin(2x 3 9x), find f (x) 24) Find the equation of the normal line to f(x) = ( x)(4 x 2 ) at x = 1. 25) If f(2) = 8, f (2) = 3, g(2) = 12 and g (2) = 4, determine the value of (fg) (2):

8 APPLICATIONS OF DERIVATIVES Determine the location (x coordinate) of the absolute maximum and minimum on the given interval: 1) f(x) = x 3 6x 2 9x + 3, from [ 3, 1] 2) f(x) = x2 3x 6, from [3, 7] For questions #3-6, use the following function: y = 3x 5 5x 3 3) Determine x and y intercept(s) for the function. 4) Determine the intervals where the function is increasing or decreasing. 5) Determine the coordinates of all relative maximums and minimums. 6) Determine the location (x coordinate) of all inflection points.

9 The position of a particle (in inches) moving along the x axis after t seconds is given by the following equation: s(t) = 2 5 t t4 4 3 t ) Calculate the velocity of the particle at time t. 8) At what times, if any, is the particle at rest? 9) At what times, if any, does the particle change direction? 10) Calculate the acceleration of the particle at time t. For questions #11-14, a calculator is allowed: 11) An open top box with a square bottom and rectangular sides is to have a volume of 32 cubic inches. Find the dimensions that require the minimum amount of material.

10 12) A farmer has 500ft of fencing to make a rectangular pen with three interior parallel partitions. What dimensions will maximize the area of the entire pen? 13) Two people on bikes are at the same place. One of the bikers starts riding directly north at a rate of 8 m/sec. Five seconds after the first biker started riding north the second starts to ride directly east at a rate of 5 m/sec. At what rate is the distance between the two riders increasing 20 seconds after the second person started riding? 14) A tank in the shape of an inverted cone is being filled with water at a rate of 12 ft 3 /sec. The base radius of the tank is 24 feet and the height of the tank is 8 feet. At what rate is the depth of the water in the tank changing when the radius of the top of the water is 12 feet?

11 INTEGRALS AND APPLICATIONS Compute the following antiderivatives using any method: 1) 2x 2 + 3x 3 + 4x 4 dx 2) t 3 t dt 3) 1 dx 4) x4 5y y dy 5) csc 2 x dx 6) 2 sin θ dθ 7) 2x2 + 3x + 4x 4 x 2 dx 8) 2e x dx 9) 1 x 2 +1 dx 10) eπ dx 2 11) (1 + tan x) dx 12) x x + 1 x x dx

12 Use the Second Fundamental Theorem of Calculus to find the result of the following: 13) d x (t 2 2t)dt dx 2 14) d dx x 2 π t cos t dt A particle moves along the x axis with the velocity given by v(t) = 2t sin(t 2 ) for t 0. 15) Find an equation for the position of the particle, given that v(0) = 2 16) Find the total distance the particle travels between t = 0 and t = π For problems #17-18, Set Up But Do Not Evaluate the Integral. 17) Find the area bounded by the curves = sin x, y = x 2 + 2, x = 1 and x = 2 18) Find the area bounded by the curves = 4x + 3, y = 6 x 2x 2, x = 4 and x = 0

13 Determine the volume of the solid of revolution by the method of disks/washers: 19) The region in Quadrant I, bounded by f(x) = 1 x 2, the x-axis, and the y-axis, rotated around the x- axis. 20) The region in Quadrant I, bounded by f(x) = 6 2x, the x axis, and the y axis, rotated around the y- axis. Set Up But Do Not Evaluate the Integral. Determine the volume of the solid of revolution by the method of cylindrical shells: 21) The region enclosed by y = x 2, x = 1, and the x-axis, rotated around the y-axis.

14 22) The region bounded by y = x 3, y = 8, and x = 0, rotated around the x-axis. Set Up But Do Not Evaluate the Integral. Determine the volume of the solid: 23) The solid has a base that is the circle x 2 + y 2 = 5, and whose cross sections perpendicular to the x axis are equilateral triangles. Set Up But Do Not Evaluate the Integral.

15 DIFFERENTIAL EQUATIONS Consider the differential equation dy = 3x2 dx e y 1) Find the general solution to the differential equation (y in terms of x). 2) Find a particular solution that satisfies the condition y(1) = 1 Consider the differential equation dy dx = y2 x 3 3) Find the general solution to the differential equation (y in terms of x). 4) Find a particular solution that satisfies the condition y(1) = 5

16 For questions #5-6, a calculator is allowed: 5) The rate of growth of the volume of a sphere is proportional to its volume. If the volume of the sphere is initially 36π ft 3, and expands to 90πft 3 after 1 second, find the volume of the sphere after 3 seconds. 6) A radioactive element decays exponentially in proportion to its mass. One-half of its original amount remains after 5750 years. If 10,000 grams of the element are present initially, how long until 1,000 grams are left?