15 calc questions 1

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1.

If f(x) is differentiable for the closed interval [-3, 2] such that f(-3) = 4 and f(2) = 4, then there exists a value c, -3 < c < 2 such that (4 points)

	f(c) = 0
	f ‘(c) = 0
	f (c) = 5
	f ‘(c) = 5

2.

If f(x) = ι(x² – 8)ι, how many numbers in the interval 0 ≤ x ≤ 2.5 satisfy the conclusion of the mean value theorem? (4 points)

	Three
	Two
	One
	None

3.

Find the exact value of the following limit: . (4 points)

4.

For which pair of functions f(x) and g(x) below will the ? (4 points)

	f(x) = ex; g(x) = x3
	f(x) = x5; g(x) = ex
	f(x) = x3; g(x) = ln(x)
	f(x) = x-2; g(x) = e-x

5.

Evaluate . (4 points)

	1
	e
	-e
	0

1.

Which of the following is the best linear approximation for f(x) = tan(x) near ? (4 points)

	2y = 4x – 3π – 2
	2y = 2x – 3π – 1
	2y = 4x – 3π – 1
	2y = x – 3π – 2

2.

If f(2) = 10 and , which of the following is the best approximation for f(2.03) using local linearization? (4 points)

	-7.97
	8.03
	10.03
	3.03

3.

The function’s value will always be greater than or equal to the local linear approximation of a function f if, for all x in an interval containing the point of tangency, (4 points)

	f “(x) < 0
	f “(x) > 0
	f ‘(x) < 0
	f ‘(x) > 0

4.

At the point of tangency, the function’s value and the local linear approximation of a function have the same value. (4 points)

	True
	False

5.

If f is a function such that , then which of the following statements must be true? (4 points)

	f(a) = 7
	f(x) is continuous at x = 7
	the linear approximation for f(x) at x = a is 7
	f ‘(a) = 7

Last 5 must SHOW ALL WORK

Let g be a function that is defined for all x, x ≠ 2, such that g(3) = 4 and the derivative of g is
g′(x) = with x ≠ 2.

1. Find all values of x where the graph of g has a critical value.
2. For each critical value, state whether the graph of g has a local maximum, local minimum, or neither. You must justify your answers with a complete sentence.
3. On what intervals is the graph of g concave down? Justify your answer.
4. Write an equation for the tangent line to the graph of g at the point where x = 3.
5. Does this tangent line lie above or below the graph at this point? Justify your answer.

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