Student s Printed Name: _Key_& Grading Guidelines CUID: Instructor: Section # : You are not permitted to use a calculator on any part of this test. You are not allowed to use any textbook, notes, cell phone, laptop, PDA, or any technology on any part of this test. All devices must be turned off while you are in the testing room. During this test, any communication with any person (other than the instructor or his designated proctor) in any form, including written, signed, verbal, or digital, is understood to be a violation of academic integrity. No part of this test may be removed from the testing room. Read each question very carefully. In order to receive full credit, you must: 1. Show legible and logical (relevant) justification which supports your final answer.. Use complete and correct mathematical notation.. Include proper units, if necessary.. Give exact numerical values whenever possible. You have 90 minutes to complete the entire test. On my honor, I have neither given nor received inappropriate or unauthorized information at any time before or during this test. Student s Signature: Do not write below this line. Problem Possible Earned Problem Possible 1a 5 6 6 1b 5 7 5 8 5 a 6 9 6 b 6 10a c 6 10b d 6 11 6 7 1 6 5a 6 1 5 Earned 5b Test Total 100 Page 1 of 8
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 exact answers. You are NOT permitted to use a calculator on any portion of this test. 1. (5 pts. each) Find the derivative of the following functions. Use appropriate notation to denote the derivative. DO NOT SIMPLIFY. x x+ e a. f( x) = x 1 xe ( 1 xex )( + e ) f ( x) x ( x + e )( x xe x e ) x = 1 xe x b. h(x) = ( x 6) 8 1 ( + x) h (( ) 8 + x) ( x) = 1 x 6 ( ). ( pts.) Given the graph of f x 11 ( 8( x 6) 7 x + ) sketch the graph of the derivative of f x ( ) shown below, ( ) on the second axes. Awarded Keep bottom 0.5 Correct derivative of top 1.5 Correct derivative of bottom w product rule Hold top 0.5 Denominator squared 0.5 - incorrectly labeling derivative or not labeling derivative -5 f /g - top of quotient rule backwards - denominator squared completely missing -1 missing minus sign for product rule -0.5 to -1 for missing parentheses simplified answers were not checked for correctness Awarded Derivative of outside, keep the inside 1 Derivative of inside Derivative of second inside 1 - incorrectly labeling derivative or not labeling derivative -5 for derivative of outside at derivative of inside -0.5 missing ( ) -5 (a+b)^n=a^n+b^n simplified answers were not checked for correctness 1 1 1 1 Awarded Constant positive derivative on left 1 Linear or cubic derivative on right 1 Open circles for not differentiable at 0 1 Correct x-intercept at correlating to min 1 do not worry about correct y-intercepts -0.5 to -1 extra points of non-differentiability Page of 8
. (6 pts. each) Find the derivative of the following functions. Use appropriate notation to denote the derivative. Simplify by combining like terms, reducing fractions, and removing negative exponents from answers. a. f (x) = e x + 1 π xπ x + x π e f x ( ) = e x + 1 π π xπ 1 x 1 + x 7 0 Awarded Derivative of each term 1 each Simplify 1 - incorrectly labeling derivative or not labeling derivative -0.5 notational errors = e x + x π 1 x x 7 b. f (t) = 1 0 + t + t 101 t t f (t) = 1 0 + t + t = 1 101 t t t 101 + t0 + t t 1 f t ( ) = 101 c. y = sec 9x t 10 + ln t 0 0t 19 +1+ t = 101 t + 10 0t19 ln t 0 +1+ t ( ) y = sec 9x = sec9x y = ( sec9x) sec9x tan9x 9 = 6sec 9x tan9x Awarded Derivative of first term, simplified 1.5 Derivative of second term Derivative of third (and fourth term).5 - incorrectly labeling derivative or not labeling derivative -1 for (a+b)/a = b -1 summing three terms into fraction without finding common denominator -0.5 notational errors Awarded Derivative of outside, keep the inside Derivative of inside Derivative of second inside 0.5 Simplify 0.5 - incorrectly labeling derivative or not labeling derivative -6 for derivative of outside at derivative of inside - sec(9x^) -0.5 missing ( ) -1 incorrect angle (x instead of 9x) -1 plus sign instead of multiplication sign -1 missing/incorrect exponent on derivative of outside, keep the inside -1 negating d/dx of inside Page of 8
6 x+7 5 x d. y = e 6x+7 ( 5x )6 ( 6x + 7)5 5x y = e 5x ( ) 6x+7 0x 18 0x 5 5x = e 5x 6x+7 = 5e 5x ( 5x ) ( ) Awarded Derivative of outside, keep the inside Derivative of inside Simplify 1 - incorrectly labeling derivative or not labeling derivative - quotient rule backwards - denominator squared completely missing -6 for derivative of outside at derivative of inside -0.5 missing ( ) -0.5 copy error -0.5 work does not follow/didn't distribute negative to second term -1 for extra "stuff" -1 major arithmetic error (i.e., (6x+7)/(5x+)=6x/5x) -1 one of inner d/dx incorrect -1 canceling in denominator in answer -1 denominator squared disappears - product rule π. (7 pts.) If f (θ) = θ cosθ, find f ''. f θ f ( ) = +θ sinθ cosθ ( θ ) = θ cosθ + sinθ + sinθ = θ cosθ + 8sinθ π f = π cos π + 8sin π = π 1 + 8 = π + Awarded Product rule for first derivative (1 point per term) Product rule for second derivative (1 point per term) Derivative of second term for second 1 derivative Evaluation 1 Simplify 1 - incorrectly labeling derivative or not labeling derivative - EACH f *g -1 missing plus sign for product rule -0.5 slight multiplication error in simplification -0.5 for copy error -7 if no product rule at all - if first derivative correct, but then rewritten incorrectly so that second derivative is wrong (i.e., work was not followed from incorrect rewrite to second derivative; work was followed for the evaluation and simplification) -0.5-1 simplifying incorrectly (depending on severity) -0.5 notation mistake (i.e., f ' ' = f ' ' (a), mixing x and theta, etc.) -0.5 missing ( ) Page of 8
5. a. (6 pts.) Let f x f (x) = f x ( ) = lim 1 x +1. ( ) = x +1. Use the limit definition of the derivative to show that x + h +1 x +1 h 0 h ( ) ( ) x + h +1 x +1 =lim h 0 h x + h +1 + x +1 =lim h 0 h ( ) h x + h +1 + x +1 x + h +1 + x +1 x + h +1 + x +1 =lim h 0 x + h +1 + x +1 = x +1 = 1 x +1 b. ( pts.) Find an equation of the line tangent to the graph of ( ) = x +1 at x = 1. ( ) = +1 = 5 = 5 f x f 1 f ( 1) = 1 +1 = 1 5 so equation of tangent to f x y 5 = 1 ( 5 x 1 ) ( ) at x = 1 is Awarded Set up 1 Conjugate 1 Simplify numerator Cancel 1 Correct evaluation 1-1 including 0/0 in work -1 missing equals -1 missing limits -0.5 limit notation carried too far up to -1 for notational errors (such as omitting the parentheses in the denominator or poor limit notation) -1 for stating the conjugate correctly but not multiplying the denominator by the conjugate (- total if remainder of problem is correct, additional -1 for incorrect solution) - Bad cancellation -6 using derivative in place of f(x) Awarded Function evaluation 1 Derivative evaluation 0.5 Equation of line 1.5-1 poor notation - tangent line is not linear -0.5 arithmetic error -1 slope incorrect no follow through from a to b because we gave f Page 5 of 8
6. (6 pts.) Find the x -values of the point(s) where the tangent line to y = 1 x is parallel to the line y = x. y = 1 x = x 1 y = 1 x = 1 x the desired slope is 1 = x 8x = 1 x = 1 8 Awarded Correct derivative Derivative = proper slope 1 Solves for x -0.5 to -1 algebra mistakes or not simplifying answer -6 setting function = slope - derivative = line - derivative = 0 only received last point of the points for solving if correct x = 1 1 8 = 8 = 1 = 1 7. (5 pts.) Find the limit. Step by step work MUST be shown. You will not be given any credit for using L Hopital s Rule. 5sint lim = 5 Awarded t 0 7t 7 lim sint t 0 t = 5 7 lim sint t 0 t = 15 15 1= 7 7 ( )( x), f ( x) = x, and g(x) is a continuous and differentiable ( ) =. Find H ( 5). ( f o g) ( x) = f ( g( x) ) = g( x) ( x) = g( x) g ( x ) ( ) = g( 5) g ( 5 ) = ( ) = 9 16 = 1 8. (5 pts.) Suppose H(x) = f o g function where g(5) = and g 5 H(x) = H H 5 Proper constants separated from sine 1 Multiplication by 1 1 Placing k/k where needed 1 Limit -5 sinkt=ksint -5 (sinkt)/t = sink -0.5 to -1 algebra mistakes or not simplifying answer only received last point of the points for limit if correct Awarded Derivative (1.5 each part ) Evaluation 1 Correct answer 1 - derivative of outside at derivative of inside H (x) can be implied -1 for H (x)=h (a) -0.5 incorrect evaluation Page 6 of 8
9. (6 pts.) Use the following table to find d dx ( ) f g( x +1) x 1 5 f( x ) - 5 1 f '( x ) 5-1 gx ( ) - 5 1 g'( x ) - 1 5 x=1. ( ( )) y = f g x +1 y = f ( g( x +1) ) g ( x +1) y ( 1) = f ( g( ) ) g ( ) = f ( 5) g ( ) = = 10. Consider the graph of f in the figure below. Awarded First chain rule Second chain rule 1 Evaluation 1 Correct answer 1 - derivative of outside at derivative of inside y can be implied -1 for y (x)=y (a) -0.5 incorrect evaluation -1 f=f 1 1 a. ( pts.) List the values of x at which f is not differentiable. For each x -value where f is not differentiable, state the reason why not. x =, 1, are sharp corners ½ pt per point and per reason, -1/ each extra pt. x = is not continuous b. ( pts.) Using the graph above, put the following values in increasing order. f f 0 ( ). ( ) = 0 ( ) is negative ( ) is positive ( ) < f ( ) < f ( 1.5) and f 1.5 f f 0 f 1.5 so f 0 right or wrong ( ), ( ), Page 7 of 8
11. (6 pts.) Find the x -values of the point(s) where the graph of f (x) = x + sin x has horizontal tangent lines on the interval [0, π ]. f x ( ) = 1+ cos x horizontal tangents when derivative is zero 1+ cos x = 0 cos x = 1 cos x = 1 x = π, π Awarded Correct derivative Derivative = 0 1 Solves for x -0.5 to -1 algebra mistakes or not simplifying answer -6 setting function = 0 only received last point of the points for solving if correct -0.5 each extra solution -1 one incorrect final answer -1.5 only presenting one solution - both solutions incorrect but process was right up till then 1. (6 pts.) Find an equation of the normal line to y = x + tan x at x = π. ( ) = π + tanπ = π + 0 = π ( x) = 1+ sec x y π y ( ) = 1+ sec π = 1+ y π 1 cosπ m nor = 1 so equation of normal to y at x = π is y π = 1 ( x π ) = 1+ ( 1) = Awarded Function evaluation 1 Derivative 1.5 Evaluation at given x value 1 Slope of normal (can be implied 1 in equation) Equation of line 1.5-1 poor notation - incorrectly labeling derivative or not labeling derivative -1 incorrect sign on slope of normal line -0.5 arithmetic error - normal line is not linear -1 y(x)=y(pi) -1 y (x)=y (pi) 1. (5 pts.) The equation of motion of a particle is s(t) = t 1t, where s is in meters and t is in seconds. Find the acceleration when the velocity is 0. Be sure to include units on your answer. s(t) = t 1t v(t) = t 1 a(t) = 6t velocity is zero when t 1 = 0 a( ) = 6 = 1 m/s t = 1 t = t =, Awarded Velocity function 0.5 Acceleration function 0.5 Velocity = 0 1 Solve 1.5 Evaluation 1 Correct units 0.5-0.5 for missing or incorrect units -1 never shows negative solution (and thus does not eliminate) -1 per instance of notation equating s and v, or v and a -0.5 notation equating a(t) and a(k) - answers without supporting work Page 8 of 8