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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

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