Mohammed Yaaseen Gomdah MATH 203 Winter 2016 F WeBWorK assignment number Practice Problems Midterm W14 is due : 03/04/2016 at 03:00am EST. The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information. This le is /conf/snippets/setHeader.pg you can use it as a model for creating les which introduce each problem set. The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind of mistake. Usually you can attempt a problem as many times as you want before the due date. However, if you are having trouble guring out your error, you should consult the book, or ask a fellow student, one of the TA's or your professor for help. Don't spend a lot of time guessing - it's not very efcient or effective. Give 4 or 5 signicant digits for (oating point) numerical answers. For most problems when entering numerical answers, you can if you wish enter elementary expressions such as 2 3 instead of 8, sin(3 pi/2)instead of -1, e (ln(2)) instead of 2, (2 + tan(3)) (4 sin(5)) 6 7/8 instead of 27620.3413, etc. Here's the list of the functions which WeBWorK understands. You can use the Feedback button on each problem page to send e-mail to the professors. 1. (1 pt) If f (x) = 3 x3 , 5. (1 pt) nd f (4). Let f (x) = Use this to nd the equation of the tangent line to the curve 3 y = x3 at the point (4, 3.00000). The equation of this tangent line can be written in the form y = mx + b where m is: x7 . x+7 f (4) = 6. (1 pt) If f (x) = 5x x + x26 x , nd f (9). and where b is: 2. (1 pt) For what values of x is the tangent line of the graph 7. (1 pt) If f (x) = 7 x(x3 8 x + 5), nd f (x). of 3 2 f (x) = 2x + 9x 25x + 6 parallel to the line y = 1x 1.4 ? Enter the x values in order, smallest rst, to 4 places of accuracy: x1 = x2 = Find f (4). 3. (1 pt) Find the y-intercept of the tangent line to 1.3 y= 6 + 7x 8. (1 pt) Evaluate the limit at (4, 0.222948160685262) . lim x0 4. (1 pt) If 3 f (t) = 4 , t sin 4x 9x 9. (1 pt) If f (x) = nd f (t). nd f (x). Find f (1). Find f (3). 1 tan x 2 sec x 10. (1 pt) Let 17. (1 pt) By using known trig identities, f (x) = 12x(sin x + cos x) f (x) = f ( ) = 4 11. (1 pt) If f (x) = (3x + 5)4 , nd f (x). sin(2x) 1+cos(2x) can be written as A. csc(2x) B. tan(2x) C. tan(x) D. sec(x) E. All of the above F. None of the above 18. (1 pt) Use a sum or difference formula or a half angle formula to determine the value of the trigonometric functions. Give exact answers. Do not use decimal numbers. The answer should be a fraction or an arithmetic expression. If the answer involves a square root it should be enter as sqrt; e.g. the square root of 2 should be written as sqrt(2); sin( 7 ) = 12 sin( )= 8 cos( 12 )= 11 cos( 8 )= Find f (5). 12. (1 pt) If f (x) = sin(x2 ), nd f (x). Find f (4). 19. (1 pt) The expressions A,B,C,D, E are left hand sides of identities. The expressions 1,2,3,4,5 are right hand side of identities. Match each of the left hand sides below with the appropriate right hand side. Enter the appropriate letter (A,B,C,D, or E) in each blank. 13. (1 pt) If f (x) = sin5 x, nd f (x). A. tan(x) B. cos(x) C. sec(x) csc(x) Find f (4). 2 D. 1(cos(x)) cos(x) E. 2 sec(x) 14. (1 pt) If f (x) = cos(5x + 8), nd f (x). 1. 2. 3. 4. 5. Find f (4). 15. (1 pt) Let cos(x) 1sin(x) + 1sin(x) cos(x) sec(x) sec(x)(sin(x))2 sin(x) tan(x) sin(x) sec(x) tan(x) + cot(x) 20. (1 pt) Simplify and write the trigonometric expression in terms of sine and cosine: 7 f (x) = 3 sin(sin(x )) f (x) = 2 + tan2 x 1 = ( f (x))2 sec2 x 16. (1 pt) Let f (x) = 7ex cos x f (x) = f (x) = 2 . 24. (1 pt) 21. (1 pt) Question 4: Without using a calculator, match each exponential function with its graph. Is the function below exponential? G(t) = (5 t)4 ? ex If so, write the function in the form G(t) = abt and enter the values you nd for a and b in the indicated blanks below. If the function is NOT EXPONENTIAL, enter NONE in BOTH blanks below. a= ? e0.8x ? ex and b = ? e0.8x 22. (1 pt) Question 8: Is the function below exponential? 7x K(x) = 3 6x If so, write the function in the form K(x) = abx and enter the values you nd for a and b in the indicated blanks below. If the function is NOT EXPONENTIAL, enter NONE in BOTH blanks below. a= and b = 23. (1 pt) Suppose y0 is the y -coordinate of the point of intersection of the graphs below. Complete the statement below in order to correctly describe what happens to y0 if the value of b (in the red graph of g(t) = b(1 + s)t below) is increased, and all other quantities remain the same. (Click on graph to enlarge) 25. (1 pt) Question 8: As b increases, the value of y0 A. decreases B. increases C. remains the same If f (x) = 2x and g(x) = the function below: 11x . Find a simplied formula for x + 12 g f (x) = 26. (1 pt) Question 14: If f (x) = e8x , g(x) = 2x + 8 , and h(x) = simplied formula for the function below: x. Find a f g(x) h(x) = 27. (1 pt) Question 20: 2 Find simplied formulas if f (x) = x5/2 , g(x) = (7x5) , 9 and h(x) = tan (5x). Find values for the constants A and P in order to make the simplied expression equal to the combination of functions below: h(x) A tan (5x) = f g(x) (7x 5)P (click on image to enlarge) 3 A= 33. (1 pt) Suppose that P= f (x) = 28. (1 pt) x2 62 and g(x) = 6 x. For each function h given below, nd a formula for h(x) and the domain of h. Use interval notation for entering each domain. 5 Let f (x) = 3x4 x + 3 . x x (A) h(x) = ( f g)(x). f (x) = h(x) = Domain = 29. (1 pt) If f (x) = 5 sin x 3 + cos x (B) h(x) = (g f )(x). nd f (x). h(x) = Domain = Find f (3). (C) h(x) = ( f f )(x). h(x) = Domain = 30. (1 pt) A function f (x) is said to have a removable discontinuity at x = a if: 1. f is either not dened or not continuous at x = a. 2. f (a) could either be dened or redened so that the new function IS continuous at x = a. Let f (x) = 7 x (D) h(x) = (g g)(x). h(x) = Domain = + 6x+21 , if x = 0, 3 x(x3) 34. (1 pt) Suppose that f (x) = 7x, g(x) = 7, if x = 0 Show that f (x) has a removable discontinuity at x = 0 and determine what value for f (0) would make f (x) continuous at x = 0. . Must redene f (0) = Hint: Try combining the fractions and simplifying. The discontinuity at x = 3 is actually NOT a removable discontinuity, just in case you were wondering. x , x3 and h(x) = 3 10x. Find ( f g h)(x). ( f g h)(x) = 35. (1 pt) Evaluate the following: 1+ x lim . x3 1 + x x+3 31. (1 pt) Let f (x) = x2 +4x+4 . Use interval notation to indicate the domain of f (x). Note: You should enter your answer in interval notation. If the set is empty, enter \"\" without the quotation marks. Domain = Enter I for , -I for , and DNE if the limit does not exist. Limit = 36. (1 pt) Use continuity to evaluate lim sin(8x + sin 3x) x 32. (1 pt) Let f (x) = 3 4x and g(x) = 3x + 4x2 . Evaluate each of the following: f (9) = g(7) = f (7) + g(7) = g(9) f (7) = f (7) g(9) = f (9) = g(9) Enter I for , -I for , and DNE if the limit does not exist. Limit = 37. (1 pt) Evaluate the limit lim 5 3x. x2 (If the limit does not exist, enter \"DNE\".) Limit = 4 43. (1 pt) Find the value of the constant b that makes the following function continuous on (, ). 38. (1 pt) Evaluate the limit lim x8+ |x 8| . x8 5x 5 5x + b f (x) = (If the limit does not exist, enter \"DNE\".) Limit = if x 2 if x > 2 b= Now draw a graph of f . 39. (1 pt) Evaluate the limit 44. (1 pt) For what value of the constant c is the function f continuous on (, ) where x2 + 11x + 18 . x9 x+9 lim f (x) = (If the limit does not exist, enter \"DNE\".) Limit = c x sin 1 x if x = 0 otherwise 40. (1 pt) Evaluate the limit a3 a . a1 a2 1 Hint: Ask what is limx0 f (x)? lim 45. (1 pt) Let f (x) = 7ex/3 . (If the limit does not exist, enter \"DNE\".) Limit = f (8) (1) = 41. (1 pt) Evaluate the limit lim s6 46. (1 pt) Differentiate: 1 s 1 6 . s6 z = w3/2 (w + cew ) (If the limit does not exist, enter \"DNE\".) Limit = z = 47. (1 pt) Let f (x) = 42. (1 pt) A function f (x) is said to have a jump discontinuity at x = a if all three of the following conditions hold: f 1 (x) = 48. (1 pt) In each part, nd the value of x in simplest form. 1 (a) x = log3 27 x= (b) x = log10 4 10 x= (c) x = log10 0.001 x= 49. (1 pt) Use the laws of logarithms to rewrite the expression ln 4 xy in terms of ln x and ln y. After rewriting ln 4 xy = A ln x + B ln y (1) lim f (x) exists. xa (2) lim f (x) exists. xa+ (3) The left and right limits are not equal. Show that x2 + 5x + 3 f (x) = 16 6x + 1 4x + 7 . Find f 1 (x). 9x + 7 if x < 5 if x = 5 if x > 5 we nd A = and B = . 50. (1 pt) Solve the following equation. If necessary, enter your answer as an expression involving natural logarithms or as a decimal approximation that is correct to at least four decimal places. has a jump discontinuity at x = 5 by calculating the limit as x approaches 5 from the left and from the right. lim f (x) = x5 lim f (x) = x5+ 22x+18 = 3x10 Now draw a graph of f (x). x= Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 5