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Name

MATH 105 FINAL

Key

Please work out each of the given problems. Credit will be based on the steps that you show towards the final answer. Show your work.

PROBLEM 1 Please answer the following true or false. If false, explain why or provide a counter example. If true, explain why.

A) If f(x) is a differentiable function that passes through the origin such that f '(x) > 2 for all x, then f(5) cannot equal 10.

Solution

B) If f(x) is a continuous function such that f '(0) = 2, f '(1) = 0, and f '(2) = -3 then f(x) has a relative maximum at x = 1.

Solution

C) Suppose that h(x) = g '(x) and that f(x) and h(x) are continuous. Then if g(a) = g(b),

$ \int_a^b f(g(x)g'(x)dx = 0 $

Solution

PROBLEM 2 Find the derivative of

A. f(x) = x cos(x²)

Solution

B.                      x² - 1
           f(x) =
                          x² + 1

Solution

PROBLEM 3 Find the limit if it exist

$ \lim\limits_{x \to 1} \frac{1 - \sqrt{x}}{1 - x} $

Solution

PROBLEM 4 Evaluate the following integrals.

A) $ \int_1^3 (x - 1)\sqrt{x^2 - 2x + 6} dx $

Solution

B) $ \int (x^2 + 1)^2 + 3 dx $

Solution

PROBLEM 5 (35 Points) You have a camera that rotates automatically positioned 400m from the space shuttle launch pad. When the space shuttle is 300m from the ground the shuttle is moving at 20 meters per second. How fast should your camera rotate at that instant?

Solution

PROBLEM 6 (35 Points) Use right sums with n = 200 to approximate the area under the curve y = 2x + 1, above the x-axis between x = 4 and x = 10.

Solution

PROBLEM 7 (35 Points) Let $ \int_{1 - x^2}^{10} ln(t^2 + 1) dt $ Find F '(x).

Solution

PROBLEM 8 (35 Points) Use the limit definition of the derivative to find the derivative of

f(x) = x² - 2x

Solution

PROBLEM 9 Let

                          1
        f(x) =
                       1 - x²

Determine any relative extrema, inflection points, intervals where y = f(x) is increasing, intervals where y = f(x) is concave up, and any asymptotes. Then use this information (not you calculator!) to graph the function.

Solution

PROBLEM 10

Below is the graph of y = f(x) . Sketch the graph of y = f '(x).

$Graph of a function with a vertical asymptote in quadrant 1, then a minimum, and then a maximum in quadrant 1 then a horizontal asymptote.$

Solution

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