Find the simplified di↵erence quotient for f(x) = x2 � 4x + 1.

Math 135 Exam 1 Review

1. Given f(x) = 1

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x + 3 + 2, find limx!�3 f(x) and limx!1 f(x).

2. Evaluate

lim x!3

x2 + 8x + 15

x2 � 9

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3. Find the simplified di↵erence quotient for f(x) = x2 � 4x + 1.

4. Given f(x) = x2+x�1, find f 0(x) using f 0(x) = limh!0 f(x + h) � f(x)

h .

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5. Find the equation of the tangent line to curve of f(x) = x2 � p x at the

point (1, 0).

6. Di↵erentiate f(x) = 3×2 � x x2 � 1

.

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7. Find an equation for the tangent line to the graph of g(x) = x p 2x + 3

at the point (3, 9).

8. The position function (in meters) of a tasmanian devil is given by s(t) = 3t3�5t2+t, where t is in seconds. a) Find the velocity and acceleration function at time t. b) What is the velocity and acceleration of the tasmanian devil when t = 2? c) When is the tasmanian devil at rest?

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9. Di↵erentiate a) f(x) = (2×2 � 6)5 and b) h(x) = 3 p x3 � 1.

10. Find the second derivative for the following functions: a) g(x) = 1

x4

and b) h(x) = 3x + 3

7x � 5 .

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