1 lim. More Tutorial at. = have horizontal tangents? 1. (3 pts) For which values of x does the graph of A) 0.

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1 1. ( pts) For which values of does the graph of f ( ) = have horizontal tangents? A) = 0 B) C) = = 1 1,,0 1 1, D) =,. ( pts) Evaluate 1 lim cos. 1 π 6 A) 0 B) C) Does not eist D) 1 1 Version A KEY Page 1 of 17

2 . ( pts) The graph of function f ( ) in the figure has a vertical asymptote at = 1 and a horizontal asymptote at y = 1. Use the graph to find all values of where the function fails to be continuous. A) =, = 1, =, = B) = 1, =, = C) =, = D) = Version A KEY Page of 17

3 4. ( pts) The graph of function f ( ) in the figure has a vertical asymptote at = 1 and a horizontal asymptote at y = 1. Use the graph to find all values of at which the function is not differentiable. A) =, = 1, =, = B) = 1, =, = C) =, = D) = 5. ( pts) Find all slant asymptotes of the function A) y = 5 B) y = + 9 C) y = 1 D) y = 0 g ( ) = Version A KEY Page of 17

4 6. ( pts) Find the second derivative of 7t f() t = + 7. A) f () t = 7t B) f () t = 7t C) f () t = 14t+ 7 D) f () t = 14t + h 7. ( pts) Evaluate lim. h 0 ( h ) A) 1 B) C) 4 D) Does not eist 8. ( pts) Evaluate lim. 1 A) 1 B) 0 C) 1 D) Version A KEY Page 4 of 17

5 9. ( pts) Find the intervals on which the function A) (, 9), ( 9, ) B) (, 9), (9, ) C) (, ) π π D),,, cosθ f ( θ ) = θ + 9 is continuous. 10. ( pts) Solve for the angle θ in the equation sin θ =, where 0 θ π. 4 1 A) B) C) π 5π θ =, 6 6 π π 5π 7π θ =,,, π 5π 7π 11π θ =,,, π π D) θ =, Version A KEY Page 5 of 17

6 11. ( pts) It can be shown that f( ) 1 cos for all values of close to zero. Use these inequalities to find lim f ( ). A) 0 B) 1 C) Does not eist D) ( pts) The accompanying figure shows the graph of y = shifted to a new position. Write the equation for the new graph. Note the graph of y = appears as a dotted line. A) B) C) D) y = ( 4) y = + ( 4) y = + ( 4) y = + ( ) 4 Version A KEY Page 6 of 17

7 1. ( pts) The graph of a function is given. Choose the answer that represents the graph of its derivative. A) B) C) D) Version A KEY Page 7 of 17

8 14. ( pts) Determine the value of the constant k for which the function f() is continuous at = 8. A) k = 7 B) k = 8 C) Impossible D) k = 56, if 8 f( ) = + k, if > ( pts) Let f( ) 1 = and g ( ) =. Find ( ( )) g f. 1 A) g( f( )) = B) g( f( )) = 1 C) g( f( )) = D) g( f( )) = 16. ( pts) Evaluate A) B) 0 C) D) lim Version A KEY Page 8 of 17

9 d. d = ( pts) Use the table to evaluate [ f( ) 4 g( ) ] A) 1 B) 0 C) 0 D) f () 5 f () g() g () ( pts) Consider the position function st ( ) = 5 + tan t. Find the average velocity over the π π interval, 4 4. A) B) C) v av v av v av = = π 4 = π D) v av = 0 Version A KEY Page 9 of 17

10 Free Response. The Free Response questions will count 45% of the total grade. Read each question carefully. In order to receive full credit you must show legible and logical (relevant) justification which supports your final answer. Give answers as eact answers. You are NOT permitted to use a calculator on any portion of this test. 1. (8 pts.) Use the graph to find each of the following limits, if it eists. (1 pt. per part a h) The graph of function f ( ) in the figure has a vertical asymptote at = 1 and a horizontal asymptote at y = 1. Infinite limits should be answered with = or =, whichever is appropriate. If the limit does not eist (and cannot be answered as or ), state DNE. y = f () a. b. c. lim f( ) = e. lim f( ) = f. 1 lim f( ) = g. + 1 lim f( ) = 1 lim f( ) = + lim f( ) DNE d. lim f( ) = 0 h. lim f( ) = 1 Correctly states limits. ( each) Graded all or nothing. No deduction for missing or inappropriate use of equals, but notate errors. 8 points Version A KEY Page 10 of 17

11 . (1 pts.) Find the limit of each of the following functions, if it eists. Show all work to receive full credit. If the limit does not eist (and cannot be answered as or ), state DNE. a. (4 pts.) lim lim = lim ( + 5) ( + 5) = lim ( + 5) = 0 ( + 5) 5 Recognizes implicitly or eplicitly indeterminate form 0/0. Factors numerator. Cancels common factor. Correctly evaluates limit. Award one point total for recognizing 0/0 and the need to factor, but incorrect factoring leads to incorrect answer or leads to incorrect work to arrive at answer. Subtract 4 points for dividing by in both numerator and denominator. Subtract 4 points for irrelevant work. Subtract ½ point for each type of notation error with a maimum of deduction for all notation errors. Types of notation errors include, but are not limited to, missing or incorrect use of equals, missing or incorrect use of parentheses, missing or incorrect limit notation such as: lim =... without function inside limit, inappropriate limit notation: lim ( + 5) = lim ( 5 + 5), and the untrue statement: lim Subtract ½ point for each minor algebra, arithmetic, and/or copy error = b. (4 pts.) lim 0 4 lim = lim ( 4 + ) 4 + = lim = =4 1 Recognizes implicitly or eplicitly indeterminate form 0/0. Rationalizes the denominator by multiplying by the conjugate of the denominator over itself. Cancels common factor. Correctly evaluates limit. Award one point total for recognizing 0/0 (I.F.) and the need to rationalize the denominator, but incorrect rationalizing leads to incorrect answer or leads to incorrect work to arrive at answer. Subtract 4 points for irrelevant work. Subtract ½ point for each type of notation error with a maimum of deduction for all notation errors. Types of notation errors include, but are not limited to, missing or incorrect use of equals, missing or incorrect use of parentheses, missing or incorrect limit notation such as: lim =... without function inside limit, and the 0 0 untrue statement: lim = Subtract ½ point for each minor algebra, arithmetic, and/or copy error. Version A KEY Page 11 of 17

12 .(continued) Find the limit of each of the following functions, if it eists. Show all work to receive full credit. If the limit does not eist (and cannot be answered as or ), state DNE. c. (4 pts.) lim + 1 There are approaches to this problem. The guidelines apply to both methods. Method I lim lim lim = = = = OR OR Method II lim = lim = lim = lim lim lim + = = = = OR Recognizes implicitly or eplicitly indeterminate form /. Divides by raised to the highest power of in the denominator. Simplifies epression. Correctly evaluates limit. Award one point total for recognizing / (I.F.) and the need to divide through by raised to the highest power of in the denominator, but incorrect algebra leads to incorrect answer or leads to incorrect work to arrive at answer. Subtract points if just states the limit with no supporting work. Subtract 4 points for irrelevant work, such as rationalizing the denominator. Subtract ½ point for each type of notation error with a maimum of deduction for all notation errors. Types of notation errors include, but are not limited to, missing or incorrect use of equals, missing or incorrect use of parentheses, and missing or incorrect limit notation such as: lim =... without function inside limit. Subtract ½ point for each minor algebra, arithmetic, and/or copy error. 0 Version A KEY Page 1 of 17

13 . (8 pts.) Use the limit definition to find the derivative of (No credit given for using derivative theorems.) 8 f( ) = f( + h) f( ) f ( ) = lim = lim 7+ + h 7+ h 0 h h 0 h 8(7 + ) 8(7 + + h) ( 7+ + h)( 7+ ) ( 7+ + h)( 7+ ) = lim h 0 h h = lim h 0 h 7+ + h 7+ Writes notation for derivative: f ( ). States the formula for the limit definition of the derivative. (Okay if implied.) Substitutes f ( + h) correctly. Gets a common denominator between the two terms in the numerator. Equivalently, clears fractions in numerator and denominator by multiplying by common denominator (in numerator) over itself. Distributes correctly. Combines like terms. Cancels common factor h. Correctly evaluates limit. (No credit if this doesn t follow from work.) Subtract 8 points for not using the limit definition of the derivative. Subtract for missing limit in the definition. Subtract ½ point for labeling as m. tan h Subtract 6 points for substituting incorrectly to get ( ) lim 7 7 h f = + + = lim = 1. h 0 h h 0 h Award a maimum of points for early egregious error which leads to trivial limit problem. Subtract 8 points for irrelevant work. Subtract ½ point for each type of notation error (eclusive of errors in the definition of the derivative, such as omitting the limit) with a maimum of deduction for all notation errors. Types of notation errors include, but are not limited to, missing or incorrect use of equals, missing or incorrect use of parentheses, mislabeling the derivative as f ( ) instead of f ( ), and missing or incorrect limit notation such as: lim =... without function inside limit. 0 ( )( ) 8 h = lim h 0 h ( 7+ + h)( 7+ ) 8 8 = = ( )(7 + ) (7 + ) Subtract ½ point for each minor algebra, arithmetic, and/or copy error. Version A KEY Page 1 of 17

14 4. (5 pts.) Find the equation of the tangent line to g ( ) = + 1 at =. Use the derivative theorems. You do not need to use the limit definition of the derivative. g ( ) = g ( ) = ( ) = 4 The slope of the tangent line is 4. g( ) = ( ) + 1= 5 The point of tangency is (, 5). y 5 = 4( ( )) The equation of the tangent line is y 5 = 4( + ) or y = 4. Circle the figure which best represents the graph of g() and the tangent line to g at =. A) B) C) Finds the derivative g ( ). Finds the slope of the tangent line g ( ). Finds the y-coordinate of the point of tangency g( ). States the equation of the tangent line to g at = in any form. Circles the correct figure representing the graph of g and its tangent line at =. Subtract ½ point for each type of notation error with a maimum of deduction for all notation errors. Types of notation errors include, but are not limited to, missing or incorrect use of equals, missing or incorrect use of parentheses, mislabeling the derivative as f ( ) instead of f ( ), and missing or incorrect limit notation such as: lim =... without function inside limit. 0 Subtract ½ point for each minor algebra, arithmetic, and/or copy error. Version A KEY Page 14 of 17

15 5. (4 pts.) The figure below is a graph of the function y = f( ) on the closed interval [ 4, 4]. The graph is made of line segments joined end to end. Graph the derivative of f. ( 1, ) y = f () ( 4, 1) (, 1) y = f () (4, ) (, 0) Correctly graphs the derivative. ( per slope of each line segment) 4 points Subtract ½ point per line segment for a closed circle at an interior point up to a maimum of for the entire problem. Interior points should be open circles and endpoints (at = 4 and = 4) should be closed. However, no deduction for open circles or missing circles at the endpoints. Subtract ½ point if the graph etends beyond = 4 or = 4 or both. Subtract for connecting the line segments in the graph of the derivative with a vertical line. Subtract points if the graph of the derivative is shifted in any direction. Version A KEY Page 15 of 17

16 6. (8 pts.) Consider the function 4 f( ) =. Grading (for each part of problem 6) Subtract ½ point for each type of notation error with a maimum of deduction for all notation errors. Types of notation errors include, but are not limited to, missing or incorrect use of equals, missing or incorrect use of parentheses, and missing or incorrect limit notation such as: lim =... without function inside limit. Subtract ½ point for each minor algebra, arithmetic, and/or copy error. a. ( pts.) Show that the graph of f ( ) has a vertical asymptote at = 0 by evaluating both the left- and right-hand limits at = 0. 4 lim f( ) = lim = lim Sets up the left- and right-hand limits at = 0. (½ point per limit) Evaluates the left- and right-hand limits at = 0. ( per limit) No work is required. 0 ( ) ( + ) + = lim = SCRATCH: ( ) lim f( ) = lim = lim = SCRATCH: points b. ( pts.) The function f ( ) has a removable discontinuity at =. Define f () so that f ( ) is continuous at =. Justify your answer using limits. Sets up the limit as. Evaluates the limit. (½ point for factoring, ½ point for canceling the common factor and evaluating) States the value of f (). So, = 0 is V.A. 4 ( ) ( + ) + + lim f( ) = lim = lim = lim = = So, f () =. ( ) ½ point ½ point c. ( pts.) Use limits to show that the graph of f ( ) has a horizontal asymptote at y = 1. Must show algebraic work, not just the answer lim ( ) lim lim lim f = = = = = Sets up the limit as or as. Divides by raised to the highest power of in the denominator. Simplifies epression. Correctly evaluates limit. Subtract if just states the limit with no supporting work. So, y = 1 is H.A. ½ point ½ point Version A KEY Page 16 of 17

17 Scantron (1 pt.) My Scantron: Check to make sure your Scantron form meets the following criteria. If any of the items are NOT satisfied when your Scantron is handed in and/or when your Scantron is processed one point will be subtracted from your test total. is bubbled with firm marks so that the form can be machine read; is not damaged and has no stray marks (the form can be machine read); has 18 bubbled in answers; has MthSc 106 and my Section number written at the top; has my Instructor s name written at the top; has Test No. 1 written at the top; has Test Version A both written at the top and bubbled in below my CUID; and shows my correct CUID both written and bubbled in. Version A KEY Page 17 of 17