You may hold onto this portion of the test and work on it some more after you have completed the no calculator portion of the test.

Transcription

1 MTH 5 Winter Term 010 Test 1- Calculator Portion Name You may hold onto this portion of the test and work on it some more after you have completed the no calculator portion of the test. 1. Consider the function f ( θ ) θ + 8 =. Find the critical numbers of f. Make sure that you θ 7 present your work in a manner that is consistent with that demonstrated and discussed in class. (8 points) Test 1 Calculator Portion 1

2 1 1 z x = x x 4. Then state all local minimum and maximum points on z after performing a first derivative test (i.e. making an increasing/decreasing table). Make sure that you present your work in a manner that is consistent with that demonstrated and discussed in class. (1 points). Find the critical numbers of the function ( ) T est 1 Calculator Portion

3 3. Find the stationary numbers of the function w 3 ( x) = x x 1. Then perform a second derivative test at each stationary number and state the appropriate conclusions. Make sure that you present your work in a manner that is consistent with that demonstrated and discussed in class. (10 points) Test 1 Calculator Portion 3

4 4. Suppose that an object moved up and down subject to the velocity function s ( t) = 0 10t where s ( t) is the object s velocity (measured in m/s) and t is the amount of time that had passed since the motion began (measured in seconds). a. Suppose that the object was 100 m above the ground seconds into its motion. Find the position function for the motion. (4 points) b. What was the net displacement for the object over the time interval [ 1, 4 ]? Write a sentence explaining the meaning of this value. (4 points) c. What was the total distance travelled by the object over the time interval [ 1, 4 ]. Make sure that you clearly communicate how you came up with your answer. (4 points) 4 T est 1 Calculator Portion

5 MTH 5 Winter Term 010 Test 1- No Calculator Portion Name You may may not come back to this portion of the test once it is turned in it is in. 1. Find each general antiderivative. Make sure that you work is laid out in a manner consistent with that illustrated and discussed in class. (3 points each) dx. a. Find ( x 4) b. Find () t sec + 4sin() t dt 3. 3x 5 dx. c. Find ( ) x /5 d. Find e dx. Test 1 No Calculator Portion 1

6 . Formally establish each limit. To earn full credit you need to notate and organize your work in a manner consistent with that illustrated and discussed in class. Page 3 has been left blank if you need additional room to work this problem. (6 points each) a. lim + x 0 3x + 1 ln 3 x 4 x b. lim x x 3 sin x lim 1 + e x c. ( ) 3/ x x T est 1 No Calculator Portion

7 Test 1 No Calculator Portion 3

8 3. For each limit given in Table 1, state the form of the limit and state whether or not the limit is of indeterminate form. That s it for this question. (4 points each) Table 1: Problem 3 Limit Form of limit Indeterminate form? ln lim t + t 0 () t lim y y 1 ln( y ) πω lim cos e e ω ln( e ω) + π θ ( θ ) ( θ ) lim tan tan 4. A certain function, h, is everywhere continuous and everywhere differentiable. The only critical numbers for h are and 5. A graph of the second derivative of h is shown in Figure 1. Which is greater, h ( ) or h ( 5)? Explain how you know. (4 points) (Voice in head says draw a picture. ) Figure 1: y = h ( x) 4 T est 1 No Calculator Portion

9 5. A student, Abel, was given a function, f, and was asked to find the absolute maximum value of f over the interval [ 3,8 ]. Abel showed all of the work in the box below, all of which was correct. Abel said that f is continuous over [ 3,8 ]. Abel established that the only critical numbers of f over [ 3,8 ] are 4 and 5. Abel created the following table. Table : Graphical behavior of f Interval Sign on f Behavior of f ( 3, 4 ) positive increasing ( 4,5 ) negative decreasing ( 5,8 ) positive increasing Abel stated that f ( 4) = 1. Finally, Abel concluded that the absolute maximum value of f over [ 3,8 ] is 1. Abel was wrong! How can that be? That is, what was the flaw in the Abel s logic? Don t just write that Abel should have done this and that Explicitly show me that it is possible that Abel was wrong. (4 points) Test 1 No Calculator Portion 5

10 6. Another student, Beatrice, was given a function, g, and was asked to find the absolute maximum value of g over the interval [ 6,10 ]. Beatrice showed all of the work in the box below, all of which was correct. Beatrice said that the domain of g included [ 6,10 ]. Beatrice established that the only critical number of g over [ 6,10 ] is 8. Beatrice created the following table. Table 3: y = g( x) x y Finally, Beatrice concluded that the absolute maximum value of g over [ 6,10 ] is 15. Beatrice was wrong! How can that be? That is, what was the flaw in the Beatrice s logic? Don t just write that Beatrice should have done this and that Explicitly show me that it is possible that Beatrice was wrong. (4 points) 6 T est 1 No Calculator Portion