Calculus. Summer Assignment

Transcription

1 Summer Review Packet for All Students Enrolling in Calculus #160 Next Year. Name: To earn credit, show all necessary work to support your answer in the space provided. Calculus Summer Assignment Name This assignment will be due the first day of class. Late assignments will have ten points deducted for each day late. Show ALL WORK for ALL PROBLEMS. Credit will not be given to those answers lacking written work. Use the internet and other resources to aid you in solving difficult problems.

2 1) Find any intercepts of the equation y = (x 1) (x + 1). a) x=1, y=-1 b) x=-1, y=1 c) x=-1, x=1, y=-1 d) x=-1, x=1, y=0 e) x=1, y=0 If the image below is the graph of f(x),then f(x) is equal to: a) (3 x ) b) (3 + x ) c) (9 x ) d) (9 + x ) e) 3x + 9 3) Find the Domain and Range of the equation y = 1 x +. a) Domain: {x x 0} Range: {y y } b) Domain: {x x 0} Range: {y y is a real number} c) Domain: {x x 1 } Range: {y y } d) Domain: {x x is a real number} Range: {y y is a real number} e) Domain: {x x } Range: {y y 0} 4) Find the slope of the line passing through the points ( 7, 3 ) 8 4 (5, 1 ). 4 4 a) 3 4 b) 3 8 c) 3 16 d) 8 3 e) -8 5) Consider the line containing the point (-,-) and the slope m=, determine which of these points is also on the line. a) (-1,0) b) (0,1) c) (-,0) d) (0,0) e) (,0)

3 6) Consider the equation f(x) =sin(x). Find f( 5π 4 ). a) 4π 5 b) c) d) 1 e) 3 7) Consider the function f(x) { x +, x 1 x. Find the function s domain and Range, then calculate +, x > 1 f(0). a) f(0)= Domain: { x x is a real number} Range: {y y } b) f(0)=- Domain: { x x is a real number} Range: {y y } c) f(0)= Domain: { x x } Range: {y y is a real number} d) f(0)=- Domain: { x x } Range: {y y is a real number} e) f(0)=1 Domain: { x x is a real number} Range: {y y } 8) Solve the equation: cos(θ) sin(θ) = 1, find all solutions on the interval [0,π) a) 0 and π b) 0 and 3π c) π and 3π d) 0, π, and 3π e) 0, π, 3π, and π 9) Consider the functions f(x) = x 1 and g(x) = cos(x). Find f o g. a) sin (x) b) cos(x 1) c) sin (x) d) cos (x + 1) e) cos (x + 1) 10) Which equation matches the graph? a) y = 3x 1 b) 1 y + x + 3 = 0 3 c) y = 1 3 x + 1 d) 1 3 y x + 3 = 0 e) 3y x + 3 = 0

4 11) Which equation matches the graph? a) x y = 6 b) x + y + 6 = 0 c) y = 1 x 3 d) x = 6 y e) y = x + 6 1) Consider the function h(θ) = 5cos ( θ ). Find the function s domain and range. a) Domain: [-, ] Range: (-5,5) b) Domain: (-5,5) Range: [-, ] c) Domain: [-5,5] Range: (-, ) d) Domain: (-, ) Range: [-5,5] e) Domain: [-, ] Range: (-.5,.5) 13) Use half-angle formulas to calculate the exact value of tan ( π 8 ). a) 1 b) 1 c) 1 d) e) 1 14) Find the equation of the line that passes through the point (-1, ) and has an undefined slope. a) 0 = x 1 b) 0 = x + 1 c) y = 1 d) y = e) x = 15) Calculate the exact value of cos ( π 4 ) + cos (π 6 ). +1 a) b) c) 6 d) e) 3

5 16) Given that ( 3, 4) is a point on an odd function. Which of these points reside on the same function? a) ( 3, 4) b) (4, 3 ) c) (, 1 ) 3 4 d) (, 1 ) 3 4 e) ( 3, 4) 17) Find an equation of the line that passes through the point (-,4) and has a slope of 3 5 a) 3x + 5y 14 = 0 b) 5x + 3y 14 = 0 c) 14 = 3x 5y d) 0 = 5y 3x 6 e) 5y 3x ) Find all real solutions to the logarithmic equation: ln (x - 1) - ln (x - 1) = ln (4) a) ln (3) b) 1+ 17, 1 17 c) -, 3 d) no real solutions e) 3 19) Which of these equations is parallel to the line 5x 3y = 0 a) 3x 5y = 0 b) y = 4 5 x + 1 c) 3y 5x = 0 d) y = 5 3 x + 7 e) 3 5 x 5 3 y = 0 0) Consider the functions f(x) = x 3 and g(x) = 1 x + 1. The graphs of f and g are a) Parallel b) Perpendicular c) Co insider d) Both A, and C e) None of the above

6 1) Find the value of the constant c for the function so the function fits the data shown in the table. a) 3 b) 3 h(x) = c x x y c) 1 3 d) 3 e) None of the above ) Find all intercepts of the function y = (x + 3). a) No intercepts b) (9,0)(0,-3) c) (3,0)(0,9) d) (-3,0)(0,9) e) (9,0)(0,3) 3) Find all real solutions to the logarithmic equation e x e 3x 3 = a) ln(5) 5 b) no real solutions c) 1 d) -1, 1 e) ln(5) 4) Consider the line with an infinite slope and containing the point (-3,4). Determine which one of these points must be on the same line. a) (-,4) b) (-,6) c) (3,4) d) (-3,10) e) (3,10) 5) Express sin ( 1 ) as a co-function of a complementary angle 4 a) sin ( π+1 4 ) b) cos ( π+1 4 ) c) sin ( π 1 4 ) d) cos ( π 1 4 ) e) cos ( π 1 4 )

7 6) Find the slope of the line passing through these points (1,) (-,4). a) m = 1 b) m = 3 4 c) m = 1 d) m = 3 7) e) m = 3 Determine the Domain and Range of the function f(x) = { x + 4, x 5 (x 5), then find f(5)., x > 5 a) f(5)=0 Domain: (-, 4) (4, ) Range: (-, ) b) f(5)=3 Domain: [4, ) Range: [0, ) c) f(5)= Domain: (4, ) Range: (-, 0) (0, ) d) f(5)=3 Domain: ( 4, ) Range: (-, ) e) f(5)= 3 Domain: (, 4] Range: (-, 0) 8) Find the slope and y-intercept (if possible) of the line 6x 5y = 15. a) slope = 6, no y intercepts 5 b) slope = 6, y intercept (-3,0) 5 c) slope = 6, y intercept (0,-3) 5 d) slope = 6, no y intercepts 5 e) slope = 6, y intercept (0,-3) 5 9) Find an equation of the line that passes through the point (-1,) and has an undefined slope. a) y = b) x = 1 c) y = x + 3 d) y = 1 e) y = x 3 30) Consider the function, y = 6 x. Find all intercepts and test for symmetry. a) (0,6)(0,-6)(6,0) symmetry over origin b) (6,0)(-6,0)(0,6) symmetry over origin c) (0,6)(0,-6)(6,0) symmetric to y axis d) (6,0)(-6,0)(0,6) no symmetry e) (6,0)(-6,0)(0,6) symmetric to y axis

8 31) Write an equation of the line with the following characteristics: Point on the line (-3,4) x-intercept=(a,0) y-intercept=(0,a) a 0 a) x = y + 1 b) x + y c) y + x = 1 d) y = 1 x + e) 4x = y 1 3) Find all intercepts of the equation y = x +3x (3x+1). a) (0,0) (-3,0) b) (0,-3) c) (0,0) d) (-3,0) e) (0,0) (3,0) 33) If f(x) = x + 3, evaluate f(x + x). a) x x + 3 b) 3 x + x c) x x 3 d) x + x + 3 e) x x ) Express tan ( π ) as a co-function of the complementary angle. 6 a) cot ( π ) 6 b) cot ( π ) 3 c) tan ( π ) 3 d) tan ( π ) 6 e) sin ( π ) ) Determine if the function f(x) = x is even, odd, or neither a) Odd b) Neither c) Even d) Undefined e) Perpendicular

9 36) Write an equation perpendicular to the line x + y = 7 with the point (-3,). a) y = x + 7 b) 7 = y + x c) yx = 7 d) y = 7 x + 1 e) x 7 y = 0 37) Express csc(0.53) as a co-function of the complementary angle. a) sec ( π 0.53) b) sec ( π ) c) csc ( π 0.53) d) csc ( π ) e) csc (0.47) 38) Which of the following equations passes through the points ( 7 8, 3 4 ) and (5 4, 1 4 ). a) y = 3 x b) 3x 1y + 37 = 0 c) y = 8 37 x 3 1 d) 1y + 3 = 3x + 40 e) 1y + 9 = 3x 8 39) Which trio of points is collinear? a) (,5)(3,7)(3,9) b) (4,6)(5,7)(5,10) c) ( 3 4, 1 ) (1,1) (9 4, 3 ) d) ( 7, 3 ) 8 16 (15, 35 ) 8 16 (3, 67 ) 8 16 e) (1.076,.974)(.065,1.974)(3.407,1.997) 40) Determine if the function f(x) = sin (x)is even, odd, or neither. a) Odd b) Neither c) Even d) Undefined e) Perpendicular

10 41) Earthquake Magnitudes on the Richter scale, the magnitude R of an earthquake of intensity I is given by: R = log I I 0 where I 0 = 1 is the minimum intensity used for comparison. Find the intensity per unit of area for the following values of R. (a) R=8.4 (b) R=6.85 (c) R=9.1 4) Sketch the graph of the equation by point plotting. y = x 4

11 43) Length A right triangle is formed in the first quadrant by the x-axis, y-axis, and the line through the point (3, ). Write the length L of the hypotenuse as a function of x. (3, ) 44) Sound Intensity The relationship between the number of decibels β and the intensity of a sound I in watts per centimeter squared is given by: β = 10 log( I 10 16) Determine the intensity of a sound in watts per centimeter squared if the decibel level is ) Prove that the product of an odd function and an even function is odd.

12 46) Falling Object In an experiment, students measured the speed s (in meters per seconds) of a falling object t seconds after it was released. The results are shown in the table. t s (a) Use the regression capabilities of a graphing utility to find a linear model for the data. (b) Use a graphing utility to plot the data and graph the model. How well does the model fit the data? Explain your reasoning. (c) Use the model to estimate the speed of the object after.5 seconds. 47) Break-Even Point Find the sales necessary to break even ( R = C) if the cost C of producing x units is: C = 5.5 x + 10,000 and the revenue R for selling x units is R = 3.9x

13 48) Verify secθ + cscθ cosθ sinθ = sinθtanθ + cosθcotθ 49) Volume An open box of maximum volume is to be made from a square piece of material 4 cm on a side by cutting equal squares from the corners and turning up the sides (see figure). (a) Use the table feature of a graphing utility to complete six rows of a table. (The first two rows are shown.) Use the result to guess the maximum volume. Height, x Length and Width Volume, V 1 4-(1) 1[4 (1)] = () [4 ()] = 800 (b) Write V as a function of x, and determine its domain. (c) Use a graphing utility to graph the volume function and approximate the dimensions of the box that yield a maximum volume.

14 50) Height of a Skyscraper When a certain skyscraper is viewed from the top of a 50 foot tall building, the angle of elevation is 59. When viewed from the street next to the shorter building, the angle of elevation is 6. (a) Approximately how far apart are the two structures? (b) Approximate the height of the skyscraper to the nearest tenth of a foot.