Student s Printed Name: _ Key _&_Grading Guidelines CUID:

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1 MthSc 7 Test Spring Version A.., 6. Student s Printed Name: _ Key _&_Grading Guidelines CUID: Instructor: Section # : You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cell phone, laptop, PDA, or any technology on either portion 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 for the free response portion of the test, you must:. Show legible and logical (relevant) justification which supports your final answer.. Use complete and correct mathematical notation.. Include proper units, if necessary. 4. Give exact numerical values whenever possible. You have 9 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. Free Response Problem Possible Earned Free Response Problem Possible 4 6b 6 6c a b Free Response 6 6a 6 Multiple Choice 4 Earned Test Total Page of

2 MthSc 7 Test Spring Version A.., 6. Multiple Choice. There are 4 multiple choice questions. Each question is worth points and has one correct answer. The multiple choice problems will count as 4% of the total grade. Use a number pencil and bubble in the letter of your response on the scantron sheet for problems 4. For your own record, also circle your choice on your test since the scantron will not be returned to you. Only the responses recorded on your scantron sheet will be graded. You are NOT permitted to use a calculator on any portion of this test.. Write " ( pts.) + " using sigma notation. k + a) k= 7 k b) k= 7 k c) k= k d) k= 7 The figure shows the areas of the regions bounded by the graph of f and the x-axis. Use the graph to answer the next two questions.. ( pts.) Evaluate the integral f (x) dx. a) - b) 7 c) - d) 4. Which of the following expressions does NOT represent the total area bounded by ( pts.) the graph of f and the x-axis between x = - and x =? a) f (x)dx + f (x)dx b) f( x) dx c) f( x) dx f( x) dx d) f( x) dx Page of

3 MthSc 7 Test Spring Version A.., Below is the graph of f x = 4x +. Use geometry to find the area of the region ( pts.) bounded by f x and the x-axis on the interval,. a) b) 4 c) d). The velocities (in/sec) of a remote controlled race car moving along a dirt path for 8 seconds are given in the following table. Estimate the distance traveled by the car using a right Riemann sum approximation to the displacement on,8 ( pts.) with n = 4. Time (sec) Velocity (in/sec) a) 4 in b) 7 in c) 6 in d) 8 in 6. Estimate the area under the curve of f x = x on the interval, using a left ( pts.) Riemann sum approximation and rectangles. a) 8 b) 7 c) 6 d) 9 Page of

4 MthSc 7 Test Spring Version A.., ( pts.) Let f (x) be an even function where f (x)dx =. Let g(x) be an odd function where g(x)dx =. Find f (x) g(x) dx. a) 6 b) c) 4 d) " 8. Use a graph and area to evaluate x dx. ( pts.) a) π b) π c) π d) π 9. ( pts.) Find d dx x sin tdt. π a) xsin x b) sin 4 x x c) 4 cos4 π 4 cos4 x d) sin x sin π. ( pts.) Evaluate the integral a) 7 b) -7 c) -4 d) 4 (4 x+ ) dx. Page 4 of

5 MthSc 7 Test Spring Version A.., 6.. Find the integral ( + ) sec ( + 4 ) ( pts.) x a) tan ( 4 ) x x x dx. + x x + x + C b) tan( 4 ) x + x + C tan x + 4 x + C c) ( ) x + sec x + 4 x + C d) ( ). ( pts.) Evaluate ( ) k= k (k ). a) 4 b) c) d) Use f x dx =, f x dx = 4,. Find f x dx. ( pts.) and g x dx = 7 to answer the next two questions. a) 7 b) c) d) 7 4. ( pts.) Find g x f x dx. a) b) 9 c) 9 d) The Free Response section follows. PLEASE TURN OVER YOUR SCANTRON while you work on the Free Response questions. You are welcome to return to the multiple choice section at any time. Page of

6 MthSc 7 Test Spring Version A.., 6. Free Response. The Free Response questions will count as 6% 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 exact answers. You are NOT permitted to use a calculator on any portion of this test.. (4 pts.) Consider the graph of f x = 6 x. Estimate the area under the graph of f x on the interval, using midpoint rectangles. Draw the rectangles on the graph and use f x, not the graph, to find the function values. f ( ) + f ( 4) = ( + ) = = 4 Correct interval Correct rectangles Midpoint values. Multiply by width of.. a. ( pts.) Find the average value for the function f x = 6x on the interval,. ( 6x ) dx = x ( x) = 7 ( ) ( + ) = 4 = 4 ( 6x ) dx f = = 4 7 = Anti-derivative Evaluation Average value -. to - poor notation - additional if final answer is a function b. ( pts.) Find the x-values that satisfy the Mean Value Theorem for Integrals for f x = 6x on the interval,. 6x = 6x = 9 x = 9 6 = Function = part a Solved -. to - poor notation answer does not have to be reduced Page 6 of

7 MthSc 7 Test Spring Version A.., 6. x if x <. Let f x = x which is defined piecewise as f (x) =. x if x a. (4 pts.) Find the total area bounded by the graph of f( x ) and the x-axis between x = and x = 4. It may help to shade the associated areas on the graph but the shading will not be graded. as signed areas: 4 ( )( ) = ( )( ) = 4 ( )( ) = so total area = = 6 = b. ( pts.) Evaluate f x dx. 4 f (x)dx = 4 + = = Each area pt each Add positives for total area Add signed areas for b 4. (6 pts.) Evaluate the following integral. If you use substitution, clearly define u and du. 4 ( 4x + ) dx 4 = u du 4x + ( ) dx u = 4x + du = 4dx x = u = x = u = = u = = u = 4 Correct u, du Correct conversion including limits Anti-derivative Evaluation & simplification -. to - poor notation - integral or anti-derivative in terms of u with x- limits Page 7 of

8 MthSc 7 Test Spring Version A. 6.. Compute the definite integral (x + ) dx as the limit of a right Riemann Sum by completing parts a. e. You may need one or more of the following common sums: k = n n + ; k = n n + n + 6 ; k = n n + a. ( pt.) Let n be the number of subintervals into which the interval is to be divided. State Δx, the width of each subinterval, in terms of n. Δx = n = right or wrong n b. ( pts.) Find an expression for the height of the k-th rectangle at the right endpoint of the k-th subinterval. The expression should be in terms of k and n. x k = + k n = k n ( ) = k n f x k + = k + n c. ( pts.) Find an expression for the area of the k-th rectangle, in terms of k and n. A k = f ( x k ) Δx = k + n n = 4k + 4 part a * part b right or wrong, consistent w/ a&b n n d. (4 pts.) State the right Riemann Sum for the function on the given interval. Simplify the Riemann Sum. State your final answer in terms of n only. n 4k + 4 n n = 4 n n 4 k + n n k= k= k= ( )( n +) = 4 n n n n n = 4 ( n + )( n +) + 4 n Correctly states x k (can be implied) Correctly subs into function (simplification not required here) = 8n +n n e. ( pts.) Find the limit of the Riemann Sum (found in part d) as the number of rectangles are increased to infinity. i.e. Find the exact value of the definite integral. 8n +n + 4 lim + 4 n n = = grade based on previous work except no credit for limit of something that results in infinitiy or zero.. Correctly sets up the sum Correctly uses the algebra properties to distribute the sum - an error above made this too simple at least - if n is a specific number (severity of algebra) Page 8 of

9 MthSc 7 Test Spring Version A Find the following integrals. Be sure to combine like terms, simplify coefficients, do not leave negative exponents in the final answer, etc. If you use substitution, clearly define u and du. a. (6 pts.) x 7 + x dx x 7 + x dx = ( u 7)u du = u 7u du u = 7 + x x = u 7 = u 7u du = dx + C = ( 7 + x ) 4 ( 7 + x ) + C Correct u, du Correct conversion Anti-derivative Return to original variable. C. -. to - poor notation -. mixed x and u - integral notation remains after anti-derivative or if dropped too soon -. not simplified b. (6 pts.) x sin(x )dx x sin(x )dx = sinu du = cosu + C = cos x + C u = x du = x dx du = x dx Correct u, du Correct conversion Anti-derivative Return to original variable. C. -. to - poor notation -. mixed x and u - integral notation remains after anti-derivative or if dropped too soon -. not simplified c. ( pts.) 9x ( x x +) dx 4 9x = u 4 du = u x x + + C ( ) 4 dx u = x x + du = ( x )dx = du = ( 9x )dx = u + C ( x x +) + C Correct u, du Correct conversion Anti-derivative Return to original variable. C. -. to - poor notation -. mixed x and u - integral notation remains after anti-derivative or if dropped too soon -. not simplified Page 9 of

10 MthSc 7 Test Spring Version A (6 pts.) The velocity of an object moving along a line is given by v(t) = 6t 8 m/s. Find the total distance traveled over the interval t seconds. Be sure to include units with your answer. 6t 8 = t = ( 6t 8)dt ( 6t 8)dt = t ( 8t) = 7 4 ( ) = 7 = t ( 8t) = ( ) = 7 = total distance = 7 + = 9 m Correct split. First integral Second integral Absolute value and total Units. -. to - poor notation -4 no split but integral correct - never took anti-derivative - F(a)-F(b) 8. ( pts.) The population of a prairie dog community grows at a rate of P ( t) = t prairie dogs/month for t. The initial value of the population was P( ) =. What is the population of prairie dogs at months? Be sure to include units with your answer. P t ( ) = P(t) = t t + C P() = + + C = C = P(t) = t t + t with P() = so P() = 4 ( ) + = = 4 prairie dogs 9. ( pt.) 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. My scantron: 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 4 bubbled in answers; has MthSc 7 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. Anti-derivative Solve for C Specific anti-derivative. Proper evaluation Units. -. to - poor notation Page of

Student s Printed Name:

MthSc 17 Test Spring 1 Student s Printed Name: CUID: Instructor: Section # : You are not permitted to use a calculator on any portion of this test. You are not allowed to use any textbook, notes, cell