Calculus I Midterm Exam. eftp Summer B, July 17, 2008

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1 PRINT Name: Calculus I Midterm Exam eftp Summer B, 008 July 17, 008 General: This exam consists of two parts. A multiple choice section with 9 questions and a free response section with 7 questions. Directions: Please clear your desk of everything except for pencils and pens. The exam is closed book and you are not allowed calculators or formula sheets. Leave substantial space between you and your neighbor. Show your work on the space provided on the exam. I can provide additional scratch paper if needed. Fill in your name at the top of this page and sign the Honor Code statement below. The Honor Code: On my honor, I have neither given nor received unauthorized aid in doing this assignment. Signature 1 of 10

2 Calculus I Midterm Exam eftp Summer B, 008 Quick Response Section 1. C a ab represents (select all that apply) ( ) b as a function of C ( ) a as a function of C ( ) a as a function of b ( ) b as a function of a ( ) C as a function of a ( ) C as a function of b. Given y = f(x), which variable is the independent variable? ( ) y ( ) f ( ) x ( ) none of the above 3. How do you test if a set of ordered pairs is a function? ( ) Horizontal line test ( ) Vertical line test ( ) If there is an x and y ( ) I just know 4. Where is/are the horizontal asymptote(s) of tan 1 ( x)? ( ) y ( ) y ( ) y ( ) y n, n integers 5. Horizontal asymptotes occur when... ( ) Functions have infinite limits ( ) x, f ( x) ( ) x 3, f ( x) ( ) Analyzing limits at infinity of 10

3 Calculus I Midterm Exam eftp Summer B, 008 Quick Response Section 6. Which is/are NOT example(s) of a tangent slope of y f (x) : Check all that apply if P and Q are separate points on the curve. f ( a h) f ( a) ( ) lim ( ) h 0 h ( ) mp ( ) y x lim h0 ( ) Average rate of change ( ) m PQ y ( ) lim x0x f ( x h) h f ( x) 7. Which of the following curves have horizontal asymptotes? ( ) 4 y x ( ) y tan 1 x ( ) sin y ( ) y x x x 8. Which of the following statement(s) is/are TRUE? ( ) If lim f x g x exists, then the limit must be f(6)g(6). x6 x 8 x 8 ( ) lim lim lim x x x x x x x 4 ( ) If f is continuous at a, then f is differentiable at a. ( ) A function can have two different horizontal asymptotes. 9. State the Squeeze Theorem for the functions fx gx hx as the limit approaches a. 3 of 10

4 Calculus I Midterm Exam eftp Summer B, Given the function f ( x) x x 5 x15 (a) Where is the function f discontinuous? State the type of discontinuity at the discontinuous point(s). (b) Find the horizontal asymptote(s) of f (x). (c) Sketch the graph of f. Label all discontinuities and asymptotes. 4 of 10

5 Calculus I Midterm Exam eftp Summer B, 008. Use the Limit Laws (DO NOT direct substitute) to find the limit as a function of n. Explicitly show and indicate each limit law used. lim rn ( r ( r ) 4)( r ) 5 of 10

6 Calculus I Midterm Exam eftp Summer B, Given f ( x) x x1 (a) Determine the derivative of f(x) at any number x using the definition of a derivative. (b) Find an equation of line normal to y = f(x) that crosses through the point (, 9). 6 of 10

7 Calculus I Midterm Exam eftp Summer B, A water balloon is thrown from a building so that its height (in feet) above the ground at t seconds is given by h( t) 16t 40t100. (a) Using the definition of a derivative, find an equation for the velocity of the balloon as a function of t. (b) Find the average velocity of the balloon from t = 1 to t = 3 seconds. (c) Find the instantaneous velocity at t = 3 seconds. 7 of 10

8 Calculus I Midterm Exam eftp Summer B, In the theory of relativity, the Lorentz contraction formula L( v) L 0 1 v c expresses the length L of an object as a function of its velocity v with respect to an observer, where L 0 is the length of the object at rest and c is the speed of light in meters per second. (a) Find the inverse of the function and explain its meaning. (b) What is the velocity of a particle whose length is 5% its rest length in terms of c. 8 of 10

9 Calculus I Midterm Exam eftp Summer B, Given the function (a) Determine f 1 ( x). f ( x) log10 ( x 5) (b) Sketch f (x) and f 1 ( x) on the same plot. (c) What is the line of reflection? (d) How does the shift from the basic log function in f (x) alter f 1 ( x)? 9 of 10

10 Calculus I Midterm Exam eftp Summer B, A ship is moving at a speed of 30 km/h parallel to a straight shoreline. The ship is 6 km from the shore and it passes a lighthouse at time t = 0. t = 0 Direction of travel Lighthouse (a) Express the distance s between the lighthouse and the ship as a function of time, so that s = f(t). (b) If the ship travels at the original speed, at what time will its distance from the lighthouse be 40 km? 10 of 10