1. ( pts) The function y = is a composite function y = f( g( )). 6 + Identify the inner function u = g( ) and the outer function y = f( u). A) u = g( ) = 6+, y = f( u) = u B) u = g( ) =, y = f( u) = 6u+ C) u = g( ) = 6+, y = f( u) = D) u = g( ) = 6+, y = f( u) = u. ( pts) Does the graph of the function y = 10+ 5sin have any horizontal tangents in the interval 0 π? If so, where? A) Yes, at B) Yes, at C) Yes, at = π π 4π =, = π π =, = D) No Version A KEY Page 1 of 17
d. ( pts) Use the table to evaluate [ f ( g ) ( )] at = 1. 1 4 f ( ) 5 5 f ( ) 5 1 4 1 g( ) 7 4 4 g ( ) 5 7 A) 46 B) 50 C) 0 D) 50 4. (4 pts) If dy y = 5, then it can be shown that =. Find y d y. A) B) C) D) d y 6y 8 = 5 9y d y 5y 8 = 5 9y d y 6y 8 = 9y d y 6y 8 = 6 9y Version A KEY Page of 17
5. ( pts) The figure shows the position s (t) of a body moving along a coordinate line as a function of time t, where 0 t 10. Use the figure to determine when the body is moving in the negative direction. s (m) A) 8< t 10 sec B) < t <, 4< t < 5, 7< t < 9 sec C) 8< t < 9 sec D) < t <, 4< t < 5, 7< t < 8 sec Version A KEY Page of 17
6. ( pts) Find the slope of the normal line to y 11 + = at the point ( ) 1,. A) B) C) 7 6 D) 7. ( pts) Use the table to evaluate A) 19 16 d f( ) g( ) at =. 1 4 f ( ) 5 5 f ( ) 5 1 4 1 g ( ) 7 4 4 g ( ) 5 7 B) 11 16 C) 19 5 D) 19 5 Version A KEY Page 4 of 17
8. ( pts) Use the graphs of the first derivate of f, y = f ( ), and the second derivative of f, y = f ( ), to choose the correct graph of y = f( ). Assume f (0) = 0. y y y = f () y = f () 1 1 1 1 A) y B) y = f () y y = f () 1 1 1 1 C) y D) y = f () y y = f () 1 1 1 1 9. ( pts) Find the derivative of y = cos(sin ). dy A) sin(cos ) = B) dy cos sin = dy C) sin(sin )cos = D) dy sin = cos Version A KEY Page 5 of 17
10. (4 pts) Two people standing together at a point begin walking at the same time and the same speed, with one walking north and the other east, so that their paths form a right angle. When the two people are both 5 m from the starting point, the triangular area formed by the positions of the two people and their starting point is changing at 4 m /s. How fast is each person walking when they are both 5 m from the starting point? A) m/s 5 B) 4 m/s 5 C) 5 m/s 4 D) 8 m/s 5 11. ( pts) At time t 0, the velocity of a body moving along the s-ais is When is the body s velocity increasing? vt () = t 7t+ 6. A) t < 6 B) 7 t < C) t > 6 D) 7 t > Version A KEY Page 6 of 17
1. ( pts) Find the derivative of 7 r = 5 θ cosθ. A) B) C) D) dr dθ = 6 7θ sin θ dr 6 7 7θ cosθ θ sinθ dθ = + dr 6 7 7θ cosθ θ sinθ dθ = dr 6 7 7θ sinθ θ cosθ dθ = 1. ( pts) Use the graph to identify the -values at which local and absolute etreme values occur. y = f () A) no local maimum, local minimum at = ; no absolute maimum, no absolute minimum B) no local maimum, local minimum at = ; absolute maimum at = 1, absolute minimum at = 4 C) local maimum at = 1, local minimum at = ; absolute maimum at = 1, no absolute minimum D) local maimum at = 1, local minimum at = ; absolute maimum at = 1, absolute minimum at = 4 Version A KEY Page 7 of 17
14. ( pts) Evaluate sin. lim 0 5 A) 5 B) 15 C) 0 D) 6 5 15. ( pts) Evaluate lim 81+ sin( π sec ). π A) 9 B) 0 C) 1 D) 8 Version A KEY Page 8 of 17
d g + f at =. 16. ( pts) Use the table to evaluate [ ( ( ))] 4 f ( ) 1 f ( ) 6 g( ) 9 g ( ) 5 5 A) 5 B) 0 C) 1 D) 5 17. ( pts) Find the slope of the tangent line to 6 6 y = 64 at the point ( ), 1. A) B) C) D) 1 1 4 Version A KEY Page 9 of 17
Free Response. The Free Response questions will count 46% 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. (1 pts.) Find the derivative of each of the following functions. Do not simplify. a. (4 pts.) dy + y = ( ) ( ) ( ) [+ ] + [ ] = ( ) 4 + 4 4 + 4 4 + = = = ( ) ( ) ( ) OR + 1 y = = ( + )( ) dy 1 = [ + ]( ) + ( + ) ( ) ( ) Applies the quotient rule (or rewrites the function as a product and applies the product rule). points General form of the answer is required here. The individual terms are graded separately. Finds the derivative of the numerator (or the first function). Finds the derivative of the denominator (or the second function using the chain rule). Subtract points for not using the quotient rule or product rule. Subtract for reversing the terms in the numerator of the quotient rule. Subtract ½ point for incorrect simplification (even though simplification is not required). 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 derivative notation such as: d =... without function inside derivative operator. Subtract ½ point for each minor algebra, arithmetic, and/or copy error. Version A KEY Page 10 of 17
1.(continued) Find the derivative of each of the following functions. Do not simplify. b. (4 pts.) gt () = 7( t t+ ) 4 4 g ( t) = [7](t+ ) + 7t 4(t+ ) () { = 7(t+ ) (t+ + 8 t) = 7(t+ ) (10t+ ) } Applies the product rule. points General form of the answer is required here. The individual terms are graded separately. Finds the derivative of the first function. Finds the derivative of the second function using the chain rule. Subtract points for not using the product rule unless the function has been epanded to take the derivative. Subtract ½ point for incorrect simplification (even though simplification is not required). 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 derivative notation such as: d =... without function inside derivative operator. Subtract ½ point for each minor algebra, arithmetic, and/or copy error. c. (4 pts.) r ( θ ) = ( secθ + tanθ) 4 ( ) ( sec tan ) ( sec tan sec ) r θ = θ + θ θ θ + θ sec θ (tanθ + sec θ) = = ( secθ tanθ) 4 secθ + ( secθ + tanθ) Applies the chain rule. points General form of the answer is required here. The individual terms are graded separately. Finds the derivative of the outside function. Finds the derivative of the inside function. Subtract points for applying the chain rule incorrectly by substituting the derivative of the inside instead of multiplying by the derivative of the inside. Subtract ½ point for incorrect simplification (even though simplification is not required). 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 derivative notation such as: d =... without function inside derivative operator. Subtract ½ point for each minor algebra, arithmetic, and/or copy error. Version A KEY Page 11 of 17
. (5 pts.) Use implicit differentiation (do not solve for y) to find dy = sec(0 y). for the equation d d ( ) = [ sec(0 y) ] dy 1 = sec(0 y) tan(0 y) 0 dy 1 1 = = cos(0 y )cot(0 y ) 0sec(0 y) tan(0 y) 0 Attempts to use implicit differentiation to take the derivative of both sides with respect to either implicitly or eplicitly. Finds the derivative of the left-hand side. Finds the derivative of the right-hand side using the chain rule. points ( for the derivative of the outside function and for the derivative of the inside function) Isolates dy/ in the final answer. Attempt to follow mistakes through work making only one deduction for the original mistake. Subtract ½ point for incorrect simplification (even though simplification is not required). 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 derivative notation such as: d =... without function inside derivative operator and incorrectly having an equation inside the derivative operator d [ = sec(0 y )]. Subtract ½ point for each minor algebra, arithmetic, and/or copy error. Version A KEY Page 1 of 17
. (7 pts.) Determine the absolute etreme values of f ( ) = 1+ 1 on the interval [, 0]. State the location(s) of each absolute etreme. f = = = + ( ) 1 ( 4) ( )( ) f ( ) = 0 when =, = f ( ) always eists ( = is not a crit. pt. because it is not in the given interval.) 0 f ( ) = 1 + 1 ( ) 1( ) + 1 = 7 + 6 + 1 = 10 ( ) 1( ) + 1 = 8 + 4 + 1 = 17 abs. ma. (0) 1(0) + 1 = 1 abs. min. f has an absolute maimum value of 17 at =. f has an absolute minimum value of 1 at = 0. Finds the derivative of f. ( per correct term in derivative) points Sets the derivative equal to zero implicitly or eplicitly. Solves for in the equation f ( ) = 0. (½ point for each value of.) Calculates the function value at the endpoints of the given interval and the one critical point. Identifies the absolute etrema by filling in the blanks correctly. (½ point per value) points Attempt to follow mistakes through work making only one deduction for the original mistake. Students should show all work when solving the equation f ( ) = 0 by finding both values of and omitting the value = from subsequent calculations because it is not in the given interval. Subtract ½ point per etrema for reversing the values in the blanks. 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 derivative notation such as: d =... without function inside derivative operator and stating f ( ) = 1= 0 which incorrectly implies that the derivative is always zero. Subtract ½ point for each minor algebra, arithmetic, and/or copy error. Version A KEY Page 1 of 17
4. (10 pts.) A rectangular swimming pool measures 0 m wide and 50 m long and is the same depth everywhere. Suppose the pool is being filled through an inflow pipe that delivers water at a rate of m /min. a. (8 pts) How fast is the depth of the water in the pool increasing? Instructions: Use t for time, h for the depth of the water in the pool, and V for the volume of water in the pool. Draw a picture and label it with known constants and given variables. Identify known rates and write an equation relating the variables. Answer the question including units. dv Given dt dt. V = 0(50) h= 1500h dv dh = 1500 dt dt dv Since = m /min, = 1500 dh. So, dt dt = m /min. Find dh 0 m dh 1 dt = 500 m/min. 50 m h Draws a picture and labels it appropriately. Identifies eplicitly or implicitly the given rate of change: dv/dt. States the equation for the volume of water in the pool. points Takes the derivative of both sides of the volume formula with respect to time t to yield equation points relating the variables and rates. ( per side) Substitutes value of given rate to find dh/dt. Correctly solves for dh/dt and provides units. Subtract ½ point for missing or incorrect units. Attempt to follow mistakes through work making only one deduction for the original mistake. 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 derivative notation such as: d =... without function inside derivative dt operator and incorrectly having an equation inside the derivative operator d [ V = 1500 h ]. dt Subtract ½ point for each minor algebra, arithmetic, and/or copy error. b. ( pts) If the pool is m deep everywhere, how long does it take to fill the pool? The volume of water required to fill the pool is 0(50)() = 000 m. OR h dh dt = 1500 So, the amount of time to fill the pool is V 000 1000 dv dt = = min. = 1000 min. Finds the amount of time to fill the pool and provides units. Subtract ½ point for missing or incorrect units. Attempt to follow work from part a. points Version A KEY Page 14 of 17
5. (1 pts.) Use the first derivative of f, ( ) = 1 ( ), to complete the problem. f a. ( pts.) State the intervals on which f is increasing and decreasing. f ( ) = 0 when = 0, = f ( ) always eists f is increasing on (,0),(0,). [OR (,)] f is decreasing on (, ). f + 0 + 0 0 States the intervals of increase. States the intervals of decrease. Because of the simplicity of this problem, no work is required for full credit. Subtract for missing or incorrect identification as intervals of increase or decrease. Subtract a maimum of ½ point for notation errors. Types of notation errors include using square brackets which indicate closed intervals instead of parentheses which indicate open intervals. Subtract ½ point for each minor algebra, arithmetic, and/or copy error. b. ( pts.) For which values of does f have local etrema? Identify each as a local maimum or local minimum. f has a local maimum at =. States the -value of the local maimum and identifies it as such. Because of the simplicity of this problem, no work is required for full credit. Attempt to follow work from part a. Subtract for missing or incorrect identification as local maimum. points Version A KEY Page 15 of 17
5.(continued)Use the first derivative of f, f ( ) = 1 ( ), to complete the problem. c. (4 pts.) State the intervals on which f is concave up and concave down. f ( ) = 1 ( ) = 6 1 f ( ) = 7 6 = 6 ( ) f ( ) = 0 when = 0, = f ( ) always eists f is concave up on (0, ). f is concave down on (,0), (, ). f 0 + 0 0 Finds the second derivative of f. Solves for in the equation f ( ) = 0. (½ point for each value of.) States the intervals of concave up. States the intervals of concave down. Subtract for missing or incorrect identification as intervals of concave up or concave down. Subtract a maimum of ½ point for notation errors. Types of notation errors include using square brackets which indicate closed intervals instead of parentheses which indicate open intervals. d. ( pts.) For which values of does f have inflection points? f has inflection points at = 0 and =. States the -value of the inflection points. ( per -value) Because of the simplicity of this problem, no work is required for full credit. Attempt to follow work from part c. points e. ( pts.) Given the ordered pairs, (0, 0), (, 48), (, 81), and (4, 0), on the graph of f, sketch the graph of y = f (). y Local maimum sketched as such and inflection points show change in concavity. General shape follows correct curve. Some attempt may be made to follow work from previous parts if mistakes are due to minor errors above. Version A KEY Page 16 of 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 17 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. 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