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Math 1271

Fall 2023

Midterm Exam 1

1. Suppose  and consider the point a = 2.

(a) (15 points) Using the limit definition of the derivative, compute f′(a).

(b) (5 points) Find the equation of the tangent line at x = a.

2. (20 points) Consider the following piecewise defined function f(x) that is drawn below:

(a) (12 points) Compute  Justify your computations.

(b) (8 points) Using limits, argue why f(x) is continuous at x = , but not differentiable at .

3. (20 points) Using that (sin(x)) = cos(x) and (cos(x)) = − sin(x), show that (cot(x)) = − csc2 (x).

4. (20 points) Let f(x) be a differentiable function. Compute the derivative of g(x) = etan(x)+x3f(x).

5. (20 points) Suppose a particle moves with position

(a) (4 points) Find the position of the particle at t = 3π/4.

(b) (8 points) Find the velocity of the particle at for any time t.

(c) (8 points) Find the acceleration of the particle at for any time t.