Summer Assignment MAT 414: Calculus

Transcription

1 Summer Assignment MAT 414: Calculus

2 Calculus - Math 414 Summer Assignment Due first day of school in September Name: 1. If f ( x) = x + 1, g( x) = 3x 5 and h( x) A. f ( a+ ) x+ 1, x 1 = then find: x+ 7, x< 1 B. ( f g) ( x) C. h( 8) D. ( go f) ( x) 1 E. g ( y). Graph x+ 5y 6= 0 on the coordinate axes below.

3 3. Find the equation of the line passing through the points P ( 7,) and ( 1, 7) Q. 4. A. Write the equation of a line with undefined slope passing through ( 7,11 ). B. Write the equation of a line with slope equal 0 and passing through ( 3, 5). 5. Use any convenient method to solve the system. 7x y= 9 x+ 3y= Describe the following functions as odd, even, or neither. A. y = x 1 B. 3 y= x + x C. y cos( x) = D. xy =

4 3 7. A. What is the set of ordered pairs that would be the inverse function for {( 1,1 ), (,8 ), ( 3, 7 ), ( 4, 64 ), ( 5,15 )} B. What property must a function have in order for its inverse to be a function? C. If f ( x) = x find f 1 ( y) 3,. D. The graph of a certain function is shown below. On the same coordinate grid sketch the graph if its inverse function 8. The graph of a certain parent function ( ) y f x = is shown below. Sketch the graph of each of the transformed functions in A C.

5 4 A. y= f ( x) 3 B. y= f ( x+ 1) C. y= f ( x) f x = x 6x + 9x Describe the end behavior for the function ( ) 3

6 5 10. Which of the four major symmetries (x-axis, y-axis, y=x, origin) do the graphs and equations below have. A. B. C. D.

7 6 x + x For the following rational function y = find the equations of the x 9 vertical and horizontal asymptotes, the location of any holes in the graph. Use you calculator to graph this function and copy the results below, showing any asymptotes as dotted lines. 1. Each of the following examples has a discontinuity. Name the x value of the discontinuity s location and what kind of discontinuity it is. A. f ( x) x = x + 3 B. g( x) = x 4 x C. x, x 1 hx ( ) = x 1, x < Find a cubic equation that has 3 and 1 i as roots.

8 7 14. Sketch a graph of a fourth degree equation with exactly two different real roots The polynomial equation Px ( ) = x 4x 4x + 0x 5 does not have any rational roots. However, it does have four irrational roots. Use your calculator to find 3 decimal place (rounded) approximations for these roots. 16. Without solving each equation use the discriminant to determine the nature (number and type) of the roots. A. 3y 10y+ 1= 0 B. w + 6w+ 10= Use synthetic division to divide (3x 5x + 6x 5) by ( x + 3). Write your answer in the space provided Answer:

9 8 18. Solve 4 x x x = 0for all its roots. 19. Solve: 3 x x = Solve: 1 + x = + 4 x x x 1. Solve over the real numbers: x x+ 10 =

10 9. A. Express 14 36'54" in decimal degrees, accurate to two decimal places. B. Express in degrees, minutes, and seconds. C. Sketch an angle of 50 in standard position and name its quadrant. D. Name two angles, one positive and one negative, which are coterminal with10. E. Name the reference angle for an angle of 30 F. Convert an angle of 7 to radians. Leave your answer as a multiple of π. G. Convert an angle of 4 π 3 radians to degrees. 3. Suppose 0 θ 90 sin θ = and ( ) 4. Find: 5 A. tan ( θ ) B. cos( θ )

11 10 4. The terminal side of θ passes through the point ( 5,1). Find the following in fraction form. A. sin ( θ ) B. sec( θ ) 5. Without using your calculator find each of the following in simple radical form: A. sin(60 ) B. tan π C. cos(135 ) D. tan π 3 6. For the triangle shown, B = 31.5 o and side c = 6.1 A b c = 6.1 A. Find A, accurate to one decimal place. o 31.5 C a B B. Find side b accurate to two significant digits. C. Find side a accurate to two significant digits.

12 11 7. In triangle LAX, x= 1, l = 15, and m A= 68 o. Find side a accurate to one decimal place. L x a A l X 8. Answer each of the following as true or false: sin θ = then θ can only be 30. If ( ) 1 The basic period for the sine function is 180 The cosecant function is undefined everywhere that the sine function is 0. If tan ( θ ) = 1.414, then ( ) tan θ = sin ( θ) cos ( 90 θ) = +. The graph at the right has a period of 360. The graph at the right has an amplitude of 3. The graph at the right could be the graph of y cos( x 45 ) = +. All trigonometric functions have an amplitude. 5 1 tan Arc cos = 13 5 Arc cos ine( x) or Cos x are between 0 o and 180 o. 1 The principal values of ( )

13 1 9. The graph at right is the graph of the sine function. Which of the following is an equation for the graph? A. y = sin( θ ) B. y = sin ( θ ) C. y = sin( θ + 45) D. y = sin( θ ) 30. For the function f ( x) 4sin(3θ 60) = + find the amplitude, period, and phase shift. Amplitude = Period = Phase shift = 31. Express 3 64s 9 t 15 in simplified form, using rational exponents Simplify; leave your answer with rational exponents: ( ) ( ) ac a c Simplify; leave your answer with rational exponents: ( x y ) y

14 For a graph of the form y x = a answer the following with true of false. A. x R B. a R C. The graph of quadrants III and IV. x x y a or y a = = will either be in quadrants I and II or D. The graph at the right is of y = ( 3 x ) y a or y a or y a E. Any graph of the form ( ) the point ( 0,1 ) or ( 0, 1). x x x = = = will pass through 35. Plot y ( + 1) = x and copy your results at the right. 36. A. Write 4 3 = 81 in logarithmic form. 1 B. Write log = 3 in exponential form. 8

15 14 C. Evaluate log ( 7 ) 1 3 D. Evaluate log5 ( 5 π ) E. Evaluate log7 ( 17.34) Solve: 3log ( 4) + 4log ( 3) = log ( x) [5] Solve log ( x 3) + log ( x+ 3) = log ( 16) [3] Use common logarithms and the base change rule to evaluate log7 ( 9 ) 40. [3] Use your calculator s graphing capabilities to solve decimal places x = 6 ( x ) accurate to four

16 [5] Solve using logarithms. ( x ) ( 3 x) 7 = 5. (Answer accurate to three decimal places.) 4. [5] Solve t = e. (Answer accurate to three decimal places.) 43. [5] The population of a certain bacteria doubles every 0 minutes. If the initial population were 30 bacilli, how many would there be in 8 hours? (Hint: this is a geometric series.) 44. [4] Write 6 1 k = 1 k! in expanded form and find the sum as a fraction.