MATH 1070 Test 3 Spring 2015 Version A , 5.1, 5.2. Student s Printed Name: Key_&_Grading Guidelines CUID:

Transcription

1 MATH 00 Test Spring 05 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:. 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 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 Earned Test Total 00 Page of

2 MATH 00 Test Spring 05 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.. (8 pts.) Using n = 4 equal-width subintervals, approximate the net area of the region bounded by the graph of f (x) = x and the x-axis on the interval [-,]. Use the midpoint of each subinterval to determine the height of each rectangle in your approximation. Sketch the rectangles on the graph below and use f ( x), NOT estimates from the graph, to find the function values. n = 4 Δx = = 4 M 4 = f + f + f = = 4 4 = + f 5 Correct interval.5 Correct rectangles midpoint function values each Final answer (only if correct) 0.5 Page of

3 MATH 00 Test Spring 05. (8 pts.) Given the function f (x) = x5 + x + x, find the particular anti-derivative of f (x) where F() =. f (x) = x5 + x + F x x ( ) = x + x x F( ) = C = x 4 + C = + C = C = 6 = 8 = x ++ x 5 + C = x + x 4 x + C x Rewrite 0.5 Anti-derivative per term point each Finding value of C.5 Rewrite (only if correct) - poor notation ( ) = x + x 4 F x x 8 Page of

4 MATH 00 Test Spring 05. (5 pts.) An oil refinery is located on the north bank of a straight river that is km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying pipe is 4 hundred thousand dollars per km over land to a point P on the north bank and then 6 hundred thousand dollars per km under the river to the tanks. To minimize the cost of the pipeline, where should P be located? x: the distance between point P and a point directly across from the storage tanks (km) a. What other variable(s) do you need to define? Include units. C : cost of pipeline (hundred thousand $) b. State the equation that needs to be optimized. Make sure it is terms of only one independent variable. ( ) C(x) = 4( 6 x) + 6 x + 4 = 4 4x + 6 x + 4 c. State the domain of the equation. 0 x 6 d. Find the derivative of the equation that needs to be optimized. C ( ) ( x) = x + 4 = 4 + 6x x + 4 x variable descriptions w/ units 0.5 Cost equation ( pt per term domain derivative.5 (0.5, respectively) -0.5 notational errors -0.5 domain open instead of closed - if incorrect equation resulted in not needing chain rule in derivative STOP. You do NOT need to find the critical point(s), you do NOT need verify that you have the location of the absolute minimum nor do you need to summarize the result. Page 4 of

5 MATH 00 Test Spring Given: f ( x)dx =, f ( x)dx =, g( x)dx = 5 Answer the following: a. ( pts.) f ( x)dx = 5 f (x)dx = f (x)dx + f (x)dx = f (x)dx 5 = f (x)dx b. ( pts.) f ( x)dx = 0 right or wrong c. (5 pts.) ( g( x) f ( x) + 4)dx = g( x)dx f ( x)dx + = = = ( ) 4 dx Shows work.5 Answer (only is correct) to - poor notation - correct answer with incorrect work/justification st and nd terms each rd term Final answer (only if correct) -0.5 per missing dx up to missing more than one equals -0.5 other poor notation note: 4 dx is area of rectangle with height 4 and width ( ) Page 5 of

6 MATH 00 Test Spring A White Sox player lightly tossed a baseball straight up in the air with initial speed of 8 ft/sec and from a height of 8 ft. Recall that gravity pull is ft/sec. Be sure to include units on your final answers. a. (5 pts.) Find the height of the ball at t seconds. You must show step by step Calculus work. facts: a t ( ) = ft/sec v(0) = 8 ft/s s(0) = 8 ft v(t) = t + C v(0) = 0 + C = 8 C = 8 v(t) = t + 8 ft/s s( t) = 6t + 8t + D s(0) = D = 8 D = 8 s( t) = 6t + 8t + 8 ft general Velocity 0.5 specific Velocity general Position specific Position unit to - for algebraic errors -0.5 to - for notation errors - extra questions answered b. (5 pts.) When does the ball hit the ground? object hits ground when s(t) = 0 6t + 8t + 8 = 0 ( ) = 0 8 t t ( )( t ) = 0 8 t + t = t = sec set position = 0 Solve Eliminate neg. solution -0.5 to - for algebraic errors - didn t address the negative solution - lack of units work should be followed from (a) unless level of algebra has changed significantly -0.5 to - for notation errors -5 solving v(t)=0 - factoring equal to non-zero Page 6 of

7 MATH 00 Test Spring (6 pts.) Given the function f (x) = sec x + 5 x + e sinh x, find the most general anti- x derivative of f (x). F(x) = tan x + 5sin x + eln x cosh x + C st term nd term.5 rd term.5 4 th term +C -0.5 to - poor notation. (8 pts.) Use geometry and area to find the exact value of ( x 4)dx. Be sure to include a graph of the function appropriately shaded along with clear and logical work related to area. 4 triangle below bh ( ) f ( ) = ( 6) = 9 triangle above bh 4 ( ) f 4 ( ) = 4 ( ) = 4 4 so ( x 4)dx = = 5 Each area pt each Add signed areas Partial credit is allowed on each area with work -5 no areas shown, FTC used but not covered in class yet Page of

8 MATH 00 Test Spring (0 pts.) An editor decided that the pages of a book should have a inch margin at both the top and bottom and a ½ inch margin on each side of the page. She also stipulated that each page of the book should have an area of 50 in. Determine the dimensions of the page that will result in the maximum printed area on the page. Your work must include a written description of the rest of the variable(s) with units, the equation to be optimized, the domain, the calculus finding the location of the optimal point and verifying that the point is optimal, and a complete sentence answering the question. x: width of the page (in) y: height of the page (in) A = + 50x A und? x = 0 nid A = 0? + 50 x = 0 50 x = x = 50 x = 5 x = 5,5 Verify that this is location of absolute max. ( 0,5): A' > 0 A is increasing ( 5, ): A' < 0 A is decreasing means x = 5 is location of max this is the only critical point so it is the absolute max ( ) A: area of printing in ( )( y ) 50 = x y A = x 50 x = y 50 A = ( x ) x = 50 x 50x + = 5 x 50x domain: 0 < x variable descriptions w/ units 0.5 basic area 0.5 area in terms of one indep. variable domain (right/wrong) derivative derivative undefined 0.5 derivative = derivative = 0 solved Exclude negative 0.5 verification Second dimension conclusion w/ units (only if correct) notational errors -0.5 lacking endpoint on st derivative number line (if used) -0.5 no units in conclusion A ( x) = 00 A ( 5) < 0 x so A(x) is concave down at x = 5 so x = 5 is location of maximum this is the only critical point so x = 5 is location of absolute maximum y = 50 5 = 0 The absolute maximum printed area of the described page occurs when the page is 5 in by 0 in. Page 8 of

9 MATH 00 Test Spring ( pts.) A runner goes out for a jog and records his running speed at regular fifteen-minute intervals. He collects the following data. Use a right rectangular approximation with 5 rectangles to estimate how far the runner ran. Be sure to include units on your final answer. 0 min 5 min 0 min 45 min 60 min 5 min Time = ¼ hr = ½ hr = ¾ hr = hr =.5 hr Speed 0 mph 6 mph.5 mph mph mph R 5 = = ( ) 4 = 4 ( ) = 4 miles Adding correct speeds.5 Multiply by correct width Final answer (only if correct) Correct units left rectangles instead of right - lack of unit conversion ( ) = x + x 4 has a root between 0 and.5. ( ). ( ) = + 4 = right or wrong 0. f x a. ( pt.) Find f f b. ( pts.) Find f ( ). f ( x) = x + ( ) = + = 4 f derivative right or wrong pt., evaluation based on derivative pt. c. ( pts.) Use Newton s Method with an initial approximation x = to find x, the second approximation of the root of f x x = 4 = ( ) = x + x 4. Correct x Correct sign Correct function/derivative Page 9 of

10 MATH 00 Test Spring 05. (0 pts.) For what positive number is the sum of its reciprocal and five times its square a minimum? Your work must include the equation to be optimized, the domain, the calculus finding the location of the optimal point and verifying that the point is optimal, and a complete sentence answering the question. S = x + 5x S = x +0x S und? x = 0 nid domain: 0 < x S = 0? x +0x = 0 x = 0x 0x = x = 0 x = 0 Proper equation to optimize (.5 per term) domain (right/wrong) derivative Critical value verification.5 conclusion w/ units 0.5 Work was followed as much as possible Verify that this is location of absolute min. 0, 0 : S ' < 0 S is decreasing, 0 : S ' > 0 S is increasing means x = 5 is location of min this is the only critical point so it is the absolute min S S ( x) = + x 0 > 0 so S(x) is concave up at x = 0 so x = is location of minimum 0 this is the only critical point so x = is location of absolute minimum 0 The absolute minimum of the described sum occurs when x = 0 Page 0 of

11 MATH 00 Test Spring Compute the definite integral ( x x)dx as the limit of a right Riemann sum by 0 completing parts a. e. You may need one or more of the following common sums:!!!! i = n n +! ; i! =!!! n n + n + 6! ; i! =!!! n n +! a. ( pt.) Let n be the number of equal width subintervals into which the interval is to be divided. State Δx, the width of each subinterval, in terms of n. Δx = 4 0 = 4 right or wrong n n b. ( pt.) Find an expression for the right endpoint of the i-th subinterval, x i. x i = 0 + 4i n = 4i right or wrong n c. ( pts.) Find an expression for the height of the i-th rectangle at the right endpoint of the i-th subinterval. The expression should be in terms of i and n. f ( x i ) = 4i n 4i n = 48i 8i n right or wrong n d. ( pts.) Find an expression for the area of the i-th rectangle, in terms of i and n. f ( x i ) Δx = 48i 8i n n 4 n = 9i i n n part a * part c right or wrong, consistent w/ a&c e. (4 pts.) Use the result of part (d) to find a formula for the sum of the areas of the n rectangles. Simplify so that your final answer in terms of n only. n 9i i n n = 9 n i n i n n i= ( ) ( n + ) = 9 n n n + 6 ( n +) n + = ( ) n ( ) i= i= ( ) n n n + 6n +6 n = n + n + 6n +6 n n = 64n + 96n + 6n +6 n n f. ( pt.) Find the limit of the Riemann Sum (found in part e) as the number of rectangles are increased to infinity. i.e. Find the exact value of the definite integral. 64n + 96n + lim n n 6n +6 n = 64 6 = 48 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) -0.5 poor notation grade based on previous work except no credit for limit of something that results in infinitiy or zero; -0.5 poor notation Page of