MTH 5 Winter Term 011 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. On this portion of the test you are expected to take all derivatives and perform all nontrivial algebra and arithmetic using your calculator. Don t forget that all derivative formulas need to be completely simplified; this includes, but is not limited to, finding the completely factored form of the derivative formula. To earn full credit, your solutions need to include all relevant information and exclude any irrelevant information. You should use appropriate calculus based techniques as illustrated in class. 4/5 1. Consider the function ( ) ( ) f x = x x 4. Find the critical numbers of f, showing all of the details that go into your determination. Make sure that you present your work in a manner that is consistent with that demonstrated and discussed in class. (8 points) Test 1 Calculator Portion 1
Mr. Simonds MTH 5 Class Winter Term 011. Find the stationary numbers of the function f ( x) = x + 1 x. 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) T est 1 Calculator Portion
3. Find the inflection points on the function f ( x) x = x 1. Make sure that you present your work in a manner that is consistent with that demonstrated and discussed in class. (10 points) 4. You are tasked to create a rectangular box with a surface area of 100 cm. The base of the box is square and the top of the box is open. Let s define x to be the length of each side of the square base (cm) and h the height of the box (see Figure 1). This question continues on page 4. (10 points total) a. Write down the formula for the surface are of the box in terms of x and h and then use the fact that the surface area is 100 cm to find a formula for h in terms of x. h x Figure 1: box with open top x Test 1 Calculator Portion 3
Mr. Simonds MTH 5 Class Winter Term 011 b. Determine the volume formula for the box in terms of x; call this formula V( x ) Completely simplify this formula. c. What is the contextual domain for the volume function found in part (b)? That is, between what two values must x lie? d. The function V has only one critical number over the contextual domain. Find this number (just show the relevant algebra you don t need to write a formal presentation). Then use an appropriate calculus-based technique to verify that the volume is indeed maximized at that value of x. 4 T est 1 Calculator Portion
. Skip Problem - Winter Term 016. Do not skip problem 3! 3. For each of the following statements, circle T if the statement if true and circle F if the statement is false. In questions where I refer to the first step I mean that you can perform the operation without doing any intervening algebra what-so-ever. (6 points) a. T or F x 3 lim x 4 x 4 has indeterminate form. x b. T or F lim ( e ) x c. T or F ( e ) sin( x) 1 sin lim x x x has indeterminate form. has indeterminate form. d. T or F You can legitimately perform L Hopital s Rule (or corollary to L Hopital s rule) sin( θ ) as your first step when evaluating lim. θ 0 sin θ e. T or F You can legitimately perform L Hopital s Rule (or corollary to L Hopital s rule) 1/ x as your first step when evaluating lim xe +. f. T or F You can legitimately perform L Hopital s Rule (or corollary to L Hopital s rule) ln( t) as your first step when evaluating lim t e t e x 0 ( ) T est 1 No Calculator Portion
4. State all of the indeterminate forms for limits. For example, one of the forms is 0. (5 points) 0 5. 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. Don t forget to state the form of the limit before beginning the problem and don t forget to state the form of the limit before each and every execution of L Hopital s Rule. This problem continues on page 4. (15 points total) a. Evaluate e lim x 0 x 1 x. x Test 1 No Calculator Portion 3
x 1 b. Evaluate lim + x 1 x 1 ln x ( ). 4 T est 1 No Calculator Portion
f x = x 3x 1x+ over the interval 3 6. Find the absolute maximum value of the function ( ) [ 0,3 ]. Make sure that you show work consistent with that illustrated and discussed during class and make sure that your conclusion is clear and on point. (10 points) Test 1 No Calculator Portion 5
7. You are told exactly two things about a function, f. The first thing you are told is that f ( 0) = 4. The second thing are told is that f ( x) 5 when x < 7 =. when x > 7 You are asked to state the absolute maximum value of the function and explain your reasoning. Give as best an answer as you can to that question. Your score will depend upon both the correctness of your response and your ability to effectively communicate your reasoning. (6 points) 6 T est 1 No Calculator Portion
MTH 5 Test 3 In Class Given on March 10, 011 Name All work on this portion of the test must be done without your calculator. To earn full credit you must outline your work and show the steps in a manner consistent with that demonstrated and discussed during lecture. 1. Evaluate each improper integral making sure that you use proper notation throughout. This problem continues on page. 10 a. Evaluate, please,. (10 points) dt ( t ) 7 7 Test 3 In Class Portion 1
b. Kindly evaluate x e x dx. (10 points) T est 3 In Class Portion
. Evaluate each integral making sure that you use proper notation throughout. This problem continues on pages 4, 5 and 6. a. Evaluate 9 e 1 ( ) ln x dx. (8 points) x b. Evaluate sec ( x) ( x ) + tan 1 dx. (4 points) Test 3 In Class Portion 3
c. Evaluate ( x) sec ( ) ( x ) + tan x dx. (4 points) tan 1 3 d. Evaluate ln ( x ) dx. (8 points) 4 T est 3 In Class Portion
e. Evaluate 1 15 x 1 xdx. (10 points) 0 Test 3 In Class Portion 5
f. Evaluate e x cos( ) x dx. (8 points) 6 T est 3 In Class Portion