This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Create assignment, 57321, Homework 8, Mar 29 at 11:11 pm 1 This printout should have 25 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. The due time is Central time. CalC5c03b 51:04, calculus3, multiple choice, > 1 min, wordingvariable. 001 A function h has graph 2 2 2 2 on ( 4 , 4). If f is defined on ( 4 , 4) by f ( x ) = 1 , 4 < x < 3 , Z x 3 h ( t ) dt, 3 ≤ x < 4 , which of the following is the graph of f ? 1. 2 2 2 2 correct 2. 2 2 2 4 2 3. 2 2 2 2 4 4. 2 2 2 2 5. 2 2 2 2 Explanation: Create assignment, 57321, Homework 8, Mar 29 at 11:11 pm 2 Since f ( 3) = Z 3 3 h ( t ) dt = 0 , two of the five graphs can be eliminated im mediately. On the other hand, by the Fun damental Theorem of Calculus, f ( x ) = h ( x ) on ( 3 , 4); in particular, the critical points of f occur at the xintercepts of the graph of h . As these xintercepts occur at 1 , , 2, this eliminates a third graph. Thus the remain ing two possible graphs for f both have the same critical points and to decide which one is the graph of f we can use the first derivative test because the graph of f will have a local maximum at an xintercept of the graph of h where it changes from positive to negative values, and a local minimum at an xintercept where h changes from negative to positive val ues. Consequently, the graph of f must be 2 2 2 2 keywords: CalC5c04s 51:04, calculus3, multiple choice, > 1 min, wordingvariable. 002 The graph of f is shown in the figure 2 4 6 8 10 2 4 6 2 If the function g is defined by g ( x ) = Z x 1 f ( t ) dt, for what value of x does g ( x ) have a maxi mum? 1. x = 1 2. x = 7 3. x = 5 correct 4. x = 6 5. not enough information given 6. x = 2 . 5 Explanation: By the Fundamental theorem of calculus, if g ( x ) = Z x 1 f ( t ) dt, then g ( x ) = f ( x ). Thus the critical points of g occur at the zeros of f , i.e. , at the x intercepts of the graph of f . To determine which of these gives a local maximum of g we use the sign chart g + 1 5 7 for g . This shows that the maximum value of g occurs at x = 5 Create assignment, 57321, Homework 8, Mar 29 at 11:11 pm 3 since the sign of g changes from positive to negative at x = 5. keywords: FTC, integral, sign chart, maxi mum CalC5c08a 51:04, calculus3, multiple choice, < 1 min, wordingvariable. 003 If the function F is defined by F ( x ) = d dx ‡ Z x 5 4 t 3 dt · , determine the value of F (1). 1. F (1) = 40 2. F (1) = 5 3. F (1) = 20 correct 4. F (1) = 10 5. F (1) = 60 Explanation: By the Fundamental Theorem of Calculus, Z x 5 4 t 3 dt = h t 4 i x 5 = x 20 . In this case, F ( x ) = d dx ‡ x 20 · = 20 x 19 Consequently, F (1) = 20 ....
View
Full
Document
This note was uploaded on 03/20/2008 for the course M 408c taught by Professor Mcadam during the Spring '06 term at University of Texas at Austin.
 Spring '06
 McAdam

Click to edit the document details