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1 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections Student s Printed Name: Instructor: XID: C Section: No questions will be answered during this eam. If you consider a question to be ambiguous, state your assumptions in the margin and do the best you can to provide the correct answer. Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed to use a tetbook, notes, cell phone, computer, or any other technology on any portion of this test. All devices must be turned off and stored away while you are in the testing room. During this test, any kind of communication with any person other than the instructor or a designated proctor is understood to be a violation of academic integrity. No part of this test may be removed from the eamination room. Read each question carefully. To receive full credit for the free response portion of the test, you must: 1. Show legible, logical, and relevant justification which supports your final answer.. Use complete and correct mathematical notation. 3. Include proper units wherever appropriate.. Give answers as eact values whenever possible. You have 90 minutes to complete the entire test. Do not write below this line. Free Response Problem Possible Earned Free Response Problem Possible Earned a b. 7. (Scantron) 1 3.a. Free Response 9 3.b. Multiple Choice Test Total 100 Version B KEY Page 1 of 15

2 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections Multiple Choice: There are 17 multiple choice questions. Each question is worth 3 points and has one correct answer. The multiple choice problems will be 51% of the total grade. Circle your choice on your test paper. Any questions involving inverse trigonometric functions should be answered based on the domain restrictions for trigonometric functions used in Section Find the vertical asymptote(s) of 9 + f ( ) = e ( 1). A) 1 1 = 0, =, = 1 = C) = 0 1 = 0, =. Evaluate or simplify the epression 1 3π tan tan. A) C) π 3π π 3π Version B KEY Page of 15

3 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections The graph gives the position s(t), in feet, of an object moving along a line at time t, in seconds, over a second interval. Find the average velocity of the object over the time interval [0.5,.5]. A) v av = 60 ft/s v av = 08 ft/s C) v av = 5 ft/s v av= 10 ft/s. Analyze the end behavior of f ( ) = 3 7. State any horizontal or slant asymptotes. A) f has slant asymptote : y= 3. f has horizontal asymptote : y= 3. C) f has horizontal asymptote : y=. f has slant asymptote : y= An equation of the line tangent to the graph of f at the point (3, 5) is y = 1. Find f (3) and f (3). A) f (3) = 7 and f '(3) = 5 f (3) = 5 and f '(3) = 7 C) f (3) = and f '(3) = 5 f (3) = 5 and f '(3) = Version B KEY Page 3 of 15

4 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections Determine whether the function g is continuous at a = 5. If g has a discontinuity at a = 5, eplain why. 1 if 5 g( ) = + 9 if > 5 A) g is continuous at a = 5. g is not continuous at a = 5 because lim g( ) g(5). 5 C) g is not continuous at a = 5 because g (5) is not defined. g is not continuous at a = 5 because lim g( ) Does Not Eist The function f in the figure satisfies lim f ( ) = 6. Determine the largest value of δ > 0 satisfying 3 the statement, If 0 < 3 < δ, then f () 6 < 3. A) δ= 0 δ= C) δ= 1 δ= 3 Version B KEY Page of 15

5 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections Given the graph of the function f, select the graph of its derivative f. y = f () A) C) 9. Assume the function g satisfies the inequality 1 g( ) sin + 1 for all values of near 0. Find lim g( ). 0 A) C) lim g( ) = 1 0 lim g( ) = 0 0 lim g( ) Does Not Eist 0 lim g( ) cannot be determined. 0 Version B KEY Page 5 of 15

6 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections Sketch a graph of a function f with all of the following properties. lim f ( ) = lim f ( ) = lim f ( ) = lim f ( ) = lim f ( ) = f (3) = 1 A) C) 11. Suppose f ( ) e and g( ) 0, with g( ) < 0 as ln5. Determine f ( ) lim ln5 g ( ). A) C) f ( ) lim = 5 g( ) ln5 f ( ) = lim ln5 g ( ) f ( ) lim = 0 g( ) ln5 lim ln5 g ( ) f ( ) = Version B KEY Page 6 of 15

7 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections Assume that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the function value g 1 ( f (3)). A) C) g 1 ( f (3)) = 3 g 1 ( f (3)) = 8 g 1 ( f (3)) = g 1 ( f (3)) = Draw a right triangle to simplify the epression 1 sin cos. Assume > 0. A) C) Version B KEY Page 7 of 15

8 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections Determine the one true statement about the value of lim f ( ). a A) lim f ( ) is always found by computing f ( a). a lim f ( ) Does Not Eist if f ( a) is undefined. a C) lim f ( ) = L if and only if lim f ( ) = L and lim f ( ) = L. + a a a lim f ( ) = L if and only if f ( a) is defined and f ( a) = L. a 15. Determine the limit lim 0. If the limit does not eist, state DNE and eplain why. A) C) lim DNE because lim DNE. 0 0 lim DNE because lim = and lim = lim = 0 0 lim DNE because lim DNE Find the intervals on which the function g is continuous. ( ) g( ) = ln A) (, ) [, ) C) (, ) (, 0),(0, ] 17. Solve the equation A) k= 3ln 8 k= ln C) k= ln8 k= e 3 8 6e k =. All students received credit for this problem on both versions due to unepected results. Item analysis revealed a discrepancy between the intended testable skill and what was the most common error in reaching the answer. Version B KEY Page 8 of 15

9 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections Free Response: The Free Response questions will be 9% of the total grade. Read each question carefully. To receive full credit, you must show legible, logical, and relevant justification which supports your final answer. Give answers as eact values. Questions involving inverse trigonometric functions should be answered based on the domain restrictions in Section (8 pts.) Use the graph of f ( ) to evaluate the limits and answer the question. Infinite limits should be answered with = or =, whichever is appropriate. If the limit does not eist (and cannot be answered as or ), state DNE. a. (1 pt.) lim f ( ) 1 = 3 b. (1 pt.) lim f ( ) 1 + = c. (1 pt.) lim f ( ) 3 DNE d. (1 pt.) lim f ( ) 1 = e. (1 pt.) lim f ( ) = 1 Parts (a) (e) Evaluates limit or states correct answer (No work required.) No deduction for missing or inappropriate use of equals signs, but note errors f. (3 pts.) State all values of for which f is discontinuous. Classify each discontinuity as either removable, infinite, or jump. = 1 jump = 1 removable = 3 infinite Version B KEY Page 9 of 15

10 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections (8 pts.) Find the following limits. Show all work. Do not use L Hopital s Rule. Infinite limits should be answered with = or =, whichever is appropriate. If the limit does not eist (and cannot be answered as or ), state DNE. sin 1 lim sin 1 a. ( pts.) π sin 1 1 (sin 1) (sin 1) π + sin + 1 π = = = π 1+ 1 sin + 1 Factors denominator points Cancels common factor Substitutes and evaluates the limit Subtract ½ point for notation errors such as missing equal sign, limit without argument, failure to drop limit on substitution, etc. with a maimum of deduction for all notation errors f ( ) 0 Subtract ½ point for the untrue statement: lim = a g ( ) 0 b. ( pts.) 1 1 lim 1 lim = ( ) = = () 16 Gets a common denominator or clears fractions in the numerator points Simplifies and cancels common factor Substitutes and evaluates the limit Subtract ½ point for notation errors such as missing equal sign, limit without argument, failure to drop limit on substitution, etc. with a maimum of deduction for all notation errors f ( ) 0 Subtract ½ point for the untrue statement: lim = a g ( ) 0 Version B KEY Page 10 of 15

11 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections (8 pts.) Find the following limits. Show all work. Do not use L Hopital s Rule. Infinite limits should be answered with = or =, whichever is appropriate. If the limit does not eist (and cannot be answered as or ), state DNE. a. ( pts.) lim ( 1)( ) + ( 1) 1 1 ( )( ) 1 ( 1) ( ) = = = (1) Rationalizes the numerator points Simplifies and cancels common factor Substitutes and evaluates the limit Subtract ½ point for notation errors such as missing equal sign, limit without argument, failure to drop limit on substitution, etc. with a maimum of deduction for all notation errors f ( ) 0 Subtract ½ point for the untrue statement: lim = a g ( ) 0 b. ( pts.) lim = = OR 5 ( + 0) + 3 Takes the square root of as [OR because 0 for all ] points Cancels common factor Substitutes and evaluates the limit Subtract ½ point for notation errors such as missing equal sign, limit without argument, failure to drop limit on substitution, etc. with a maimum of deduction for all notation errors Subtract for taking the square root of as Version B KEY Page 11 of 15

12 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections (7 pts.) Use the Intermediate Value Theorem to show that the equation has a solution in the given interval = 0; ( 1,1) Let 5 f ( ) = a. ( pts.) Calculate f ( 1) and f (1). f ( 1) = f (1) = 5 ( 1) + 7( 1) + 5= = 3 5 (1) + 7(1) + 5= = 13 Evaluates f ( 1) Evaluates f (1) Subtract ½ point for notation and minor arithmetic errors b. ( pts.) Provide all reasoning why the Intermediate Value Theorem applies to this problem. (HINT: You should have at least reasons.) Since f is a polynomial, f is continuous on (, ). So, f is continuous on [ 1, 1]. The number 0 is strictly between f ( 1) and f (1). States that f is continuous on the closed interval [ 1, 1] and provides some reasoning States that the number 0 or the RHS of the equation is between the function values (Could be stated as above or in interval notation or as an inequality.) points points c. (1 pt.) State the conclusion of the Intermediate Value Theorem as it applies to this problem. If part (b) is incorrect, no credit will be awarded for part (c). By the IVT, there eists a value c in ( 1, 1) such that f (c) = 0. States the conclusion of the IVT, giving the interval in which a solution eists No deduction if IVT is not mentioned in conclusion because it is stated in the problem. Version B KEY Page 1 of 15

13 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections (7 pts.) Use the delta-epsilon definition of a limit to prove ( ) Let f ( ) = 5. lim 5 = 3. 1 Let ε > 0 be given. ε Assume 0 < 1 < δ, where δ =. 5 (Fill in the blank with your choice of δ.) (Complete your proof below.) f ( ) 3 = (5 ) 3 = 5 5 = 5( 1) = 5 1 < 5δ ε = 5 5 = ε By the definition of a limit, lim(5 ) = 3. 1 Determines a value for δ, no work required points Substitutes correct f () and value L into f () L (or into inequality) points Shows f () 3 < ε 3 points Subtract if epsilon and delta are switched, but consistent and otherwise the work is correct Award full credit for alternative proof worked in reverse beginning with assumption 1 < δ = ε/5 and manipulating to get (5 ) 3 < ε Subtract for having = where < should be Subtract points for beginning proof with f () L < ε Subtract 3 points for attempting to prove the limit by finding δ Subtract points for showing only how to find δ Work: f ( ) 3 < ε (5 ) 3 < ε 5 5 < ε ε ε 5( 1) < ε 5 1 < ε 1 < Choose δ = 5 5 Version B KEY Page 13 of 15

14 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections (10 pts.) Use the limit definition of the derivative to find f ( ) if f ( ) a b c and c are constants. (You will receive no credit for using derivative theorems to find f ( ) = + + where a, b, a. ( pts.) Complete the limit definition to find the derivative for any function f (). (That is, state the formula for finding the derivative.) f ( + h) f ( ) f ( ) h 0 h provided the limit eists. (Partial Credit was allowed in the following situations.) Subtract for using a instead of Subtract for omitting one (but not both) of the function notations f in the numerator b. (5 pts.) Calculate the limit from part (b) {should be (a)}. Show all work. ( c) a( + h) + b( + h) + c a + b + f ( ) h 0 h h 0 ( ) a + h + h + b + bh + c a b c a + ah + ah + bh a h h 0 h h 0 h 0 ( ) a + ah + b = a + a(0) + b = a + b c. (3 pts.) Find the equation of the tangent line to ( ) tan h ( ) f ( ) = (3) = 6 m = f = 6() = 1 ( a + ah + b) h Substitutes function into formula points Simplifies to a form where common factor of h can be canceled points Substitutes and evaluates limit, following work Subtract ½ point for notation errors such as missing equal sign, limit without argument, failure to drop limit on substitution, etc. with a maimum of deduction for all notation errors Subtract for algebra errors, including: ( + h) = + h f = 3 at =. Point of tangency: f () = 3() = 8 (, 8) Equation of tangent line: y 8 = 1( ) y = 1 16, all or nothing Evaluates f () to find the slope of the tangent line Evaluates f () to get the y-coordinate of the point of tangency (okay to be implicit in the equation of the tangent line) Finds an equation of the tangent line, following work from slope and pt. of tangency Subtract ½ point for notation errors such as using derivative notation when it is not appropriate, not labeling the derivative, etc. with a maimum of deduction for all notation errors.) Version B KEY Page 1 of 15

15 MATH 1060 Test 1 Fall 018 Calculus of One Variable I Version B KEY Sections 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 17 bubbled in answers; has MATH 1060 and my section number written at the top; has my instructor s last name written at the top; has Test No. 1 written at the top; has the correct test version written at the top and bubbled in below my XID; shows my correct XID both written and bubbled in; Bubble a zero for the leading C in your XID. Please read and sign the honor pledge below. On my honor, I have neither given nor received inappropriate or unauthorized information at any time before or during this test. Student s Signature: Version B KEY Page 15 of 15