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CALCULUS I Math 2104 Write on Exam Nunley Graphing Calculator allowed CALCULUS I EXAM 1 · Show all work. Answers without work or explanation will not be given full credit. 1) a) Define what it means for f(x) to be continuous at x = c. b) List three ways in which f(x) can fail to be continuous at x = c. 2) Compute the following limits: a) b) ÷ ÷ ø ö ç ç è æ - + ® 2 2 1 x x - 1 6 5x x lim c) x tan lim 2 x + ® p (Hint: Think about the graph of the tangent.) d) ÷ ÷ ø ö ç ç è æ + - + + ¥ ® 8 7 1 6 5 lim 3 3 - x x x x x 3) For the function graphed below answer the following questions. tanx) (x lim 4 x + ® p a) = - ® ) x ( f lim 2 x b) = ® ) x ( f lim 2 x c) = - ® ) x ( f lim 5 x d) What is f(x) lim x ¥ ® ? 4) Sketch graphs of functions that have each of the following properties: a) The function is continuous at every point except x = 0 and f(x) lim 0 x ® doesn’t exist. b) , f(x) lim 2 x ¥ = - ® EMBED Equation.3 ¥ = + ® f(x) lim 2 x , ¥ = ¥ ® f(x) lim x , and 3 - f(x) lim - x = ¥ ® . 5) The graph of f(x) is given below. Use it to answer the following questions: a) Approximately what is the slope of the graph at x = -2? b) What is the slope of the tangent line at x = 1? c) Is the derivative at x = 0 positive, negative, or zero? How do you know? 6) Using the definition, find the derivative of 5x - 3x f(x) 2 = . 7) Let an object’s position be given by s(t) = 2 t - t meters after t seconds. Find: a) The average velocity from t = 1 to t = 3 seconds. b) The velocity at t = 1 second. 8) Using a table, estimate ÷ ø ö ç è æ ® 1 - x ) ln( lim 1 x x to three decimal places. _1263105323.unknown _1263105643.unknown _1548654069.unknown _1548654196.unknown _1643687079.unknown _1274250122.unknown _1263105531.unknown _1263105642.unknown _1263105640.unknown _1263105419.unknown _1219052456.unknown _1232338466.unknown _1263105300.unknown _1219120850.unknown _1219052415.unknown CALCULUS I Math 2104 Nunley CALCULUS I EXAM 4 · Show all work. Answers without work or explanation will not be given full credit. 1) Compute the following integrals: a) ò + dx 12 8x - e x b) ò 1 0 3 dx x - x c) ò + dx 2) (5x sec 2 d) dx 1 x 2x 2 0 2 ò + e) ò dx (x) 3sec - tan(x) 2 f) dx ) xcos(x 0 2 ò p 2) Given that 10 dx g(x) and 4, - dx f(x) 8, dx f(x) 3 0 6 3 3 0 = = = ò ò ò , find the following: a) ò 6 0 dx f(x) b) ò + 3 0 dx 2g(x) f(x) - c) ò 0 3 dx g(x) 3) Compute the average value of x 5e f(x) = over the interval [0,6]. 4) The following table gives the velocity of an object in free fall: Time elapsed (in seconds) 0 2 4 6 8 10 Velocity (in yards per second) 0 9 25 52 63 80 Use this information (and the fact that the velocity is constantly increasing) to give upper and lower bounds for the distance that the object falls during the 10 second interval. 5) Let 2 4 ) ( x x f - - = . a. Sketch the graph of f(x) (Hint: It’s part of a circle.) b. Use the area definition to find ò - 2 2 ) ( dx x f 6) Compute dt ) cos(t dx d 3 x x 2 ò . 7) The following graph shows the growth rate of US exports (in billions of dollars per year),E(t), and the growth rate of US imports (in billions of dollars per year), I(t), from 1945 to 1975. a) What are the units on ? dt E(t) 1975 1945 ò b) What does ò 1975 1945 dt I(t) - E(t) represent? c) Based on the graphs, is ò 1975 1945 dt I(t) - E(t) positive, negative, or zero? How do you know? _1240718583.unknown _1240718814.unknown _1398146219.unknown _1398146238.unknown _1366723018.unknown _1385451492.unknown _1353906800.unknown _1240718784.unknown _1240718790.unknown _1240718774.unknown _1240718472.unknown _1240718508.unknown _1180769898.unknown _1180770042.unknown _1240718424.unknown _1180769970.unknown _1180768375.unknown CALCULUS I Math 2104 Nunley CALCULUS I EXAM 3 · Show all work. Answers without work or explanation will not be given full credit. 1) The following is the graph of dx df . Use it to answer the following questions about f(x). (If you can’t tell exactly where something occurs, give an approximation): a) Where are the critical points for f(x)? How do you know? b) Where is f(x) increasing? Decreasing? How do you know? c) Which critical points are local minimums? Which are local maximums? How do you know? d) Where does f(x) have an inflection point? How do you know? e) Where is f(x) concave up? Where is it concave down? How do you know? 2) Compute the following limits: a) 7 - 6x x 3 4x - x lim 2 2 1 x + + ® b) 3 0 x x 6x - 6sinx lim ® (Hint: You will need to use L’Hospital’s multiple times.) c) x x / 2 0 x ) 1 ( lim + ® 3) Let f(x) = 1 - 3x 5x - x 2 3 + . a) Find the critical points for f(x). b) Determine where f(x) is increasing and where it is decreasing. c) Determine where f(x) is concave up and where it is concave down. d) Where does f(x) have its inflection points? 4) Find the absolute maximum and minimum values for f(x) = x + sinx over [0,2]. 5) A 10 foot ladder is leaning against a wall. The bottom of the ladder starts to slide away from the wall at the rate of 0.1 feet per second. How fast is the top of the ladder moving when the bottom of the ladder is 3 feet from the wall? 6) At low temperatures, the pressure P and the volume V of an ideal gas are related by the formula PV = k, where k is a constant. Assume that when the pressure is 2 atmospheres and the volume is 5 cubic meters, the pressure is increasing at the rate of 0.1 atmospheres per second. Find the rate of change of the volume. 7) A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 800 m of wire at your disposal, what is the largest area that you can enclose, and what are its dimensions? _1238901976.unknown _1617686852.unknown _1617686879.unknown _1460275127.unknown _1225185891.unknown