# Liumid1au22

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MATH 124 Midterm 1 October 25, 2022 Name: Student #: Problem: 1 2 3 4 5 Total Points: 15 15 15 8 7 60 INSTRUCTIONS: You have 80 minutes to take the test. There are 5 problems. Make sure you have all of them. Write your solution below the problem. There is scratch paper at the back of the test. The test is double-sided. Make sure you are reading the backs of pages! Unless otherwise stated, show all your work for full credit . Unless otherwise stated, all answers should be exact, without rounding. You are allowed to use one 8.5" × 11" sheet of notes, front and back. You can use a TI-30X IIS calculator. No other calculator is allowed. TIPS: The number of points a question is worth is not correlated to its difficulty. Don't spend too much time on one problem if you haven't looked at the rest of the test. There is partial credit. Even if you can't fully solve a problem, explaining your progress might get you a significant number of points. Make sure your calculator is in radians!!! Good luck!
1. For each of these questions, find the limit. If the limit does not exist, state whether the limit is or -∞ if either applies; if neither applies write "DNE". (a) (5 points) lim x 1 x 2 + x - 2 x 2 - 1 (b) (5 points) lim x →∞ 3 x 2 + x + 1 7 x 2 - x - 8 (c) (5 points) lim x 0 + cos( π - x ) x Page 2
2. (a) (5 points) Let f ( x ) = sin x + cos x - x - 1 x . Find f 0 ( x ). (b) (5 points) Let f ( x ) = (4 x 2 + x ) e x . Find f 0 ( x ). (c) (5 points) Find the equation of the tangent line to the graph of y = x 3 - 3 x - 1 at the point (2 , 1). Page 3
3. For this problem, you do not have to show your work . Answer the following questions based on the graph of f ( x ) shown below. If a limit does not exist, state whether that limit is or -∞ if either applies; if neither applies write "DNE". - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 7 8 9 10 - 5 - 4 - 3 - 2 - 1 1 2 3 x f ( x ) (a) (1 point) lim x 0 - f ( x ) (b) (1 point) lim x 0 + f ( x ) (c) (1 point) lim x 0 f ( x ) (d) (2 points) lim x 8 f ( x ) (e) (2 points) List all values of a where f 0 ( a ) = 0. (f) (2 points) List all values of a where f 0 ( a ) is undefined. (g) (2 points) lim h 0 f (2 + h ) + 1 h (h) (2 points) lim x 8 f 0 ( x ) (i) (2 points) lim h 0 + f (4 + h ) - 2 h (Be careful!) Page 4
4. Let a be a number, and let f ( x ) = ( 3 ax - 3 if x < 1 ax 2 + x if x 1 (a) (4 points) Find all values of a which make f ( x ) continuous at x = 1. If there are none, explain why. (b) (4 points) Find all values of a which make f ( x ) differentiable at x = 1. If there are none, explain why. Page 5
5. A scientist is studying a function f ( x ). Through experiment, they have obtained a table of approximate values of f ( x ) and an approximate graph of f ( x ), shown below. x f ( x ) 0.0 2.3 0.5 3.0 1.0 4.0 1.5 5.2 2.0 6.7 2.5 8.4 3.0 10.4 3.5 12.6 4.0 15.1 4.5 17.8 5.0 20.8 0 1 2 3 4 5 0 5 10 15 20 x f ( x ) Using a computer, the scientist also approximates the function f 0 ( x ). The results are shown below. x f 0 ( x ) 0.0 1.2 0.5 1.7 1.0 2.2 1.5 2.7 2.0 3.2 2.5 3.7 3.0 4.2 3.5 4.7 4.0 5.2 4.5 5.7 5.0 6.2 0 1 2 3 4 5 0 2 4 6 x f 0 ( x ) (Problem 5 continued on next page.) Page 6
(a) (3 points) The scientist believes f ( x ) is either a quadratic function (that is, of the form f ( x ) = ax 2 + bx + c where a , b , and c are constants) or an exponential function (that is, of the form f ( x ) = ab x where a and b are constants). Based on the given information, which do you think is correct? Explain your answer. Hint: Think about what the derivative of a quadratic function looks like, and what the derivative of an exponential function looks like. (b) (4 points) If you think f ( x ) is a quadratic function ax 2 + bx + c , find a , b , and c . If you think f ( x ) is an exponential function ab x , find a and b . Your values do not have to be exact. Remember to show your work or explain your answers. Page 7
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