Examen
Exercises Topic 1
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- Mathematics 1
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Worksheet 2 (17-18). Differentiation-STEPHANIE REINA FriOct/6/2017, 22/13Student: _____________________ Instructor: STEPHANIE REINA Assignment: Worksheet 2 (17-18).
Date: _____________________ Course: MATHEMATICS I Differentiation
1. Here is a graph of f(x) = x2 and the line tangent to f(x) at (a,f(a)). The equation of the y
tangent line is y = − 8x − 16.
Find a, f(a), and the slope of the parabola f(x) at (a,f(a)).
x
a
a=
f(a) =
The slope of the parabola at (a,f(a)) is .
2. Find the point(s) on the graph to the right where the slope is equal y
3
to . y = x3
11
(x,y)
x
Slope is 3x2
3
The point(s) on the given graph at which the slope is equal to is(are) .
11
(Type ordered pairs. Use a comma to separate answers as needed. Type exact answers, using radicals as needed.
Use integers or fractions for any numbers in the expression.)
3. Consider the curve y = f(x) shown below. Find f (15) and f (15) =
f ′ (15).
f ′ (15) =
y y=f(x)
5
y= x+5
14
15 x
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,Worksheet 2 (17-18). Differentiation-STEPHANIE REINA FriOct/6/2017, 22/13
4. 1 3 2
The tangent line to the curve y = x − 4x + 17x + 16 is parallel to the line 8x − 4y = 3 at two points on the curve.
3
Find the two points.
The two points are .
(Type an ordered pair. Use a comma to separate answers as needed.)
5. Sketch the graph of a function that has the properties described below.
f(4) = − 2; f ′(x) < 0 for x < 4, f ′(4) = 0 and f ′(x) < 0 for x > 4.
Choose the correct graph below.
A. B. C. D.
y y y y
x x x x
(4, -2)
(4, -2) (4, -2) (4, -2)
6. Graph the derivative of the function graphed on the right. y
x
Choose the correct graph below.
A. B. C. D.
y y y y
x x x x
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, Worksheet 2 (17-18). Differentiation-STEPHANIE REINA FriOct/6/2017, 22/13
7. The graph of the accompanying figure shows the dy
average annual percentage change y = f(t) in a particular Choose the correct graph of .
dt
country's gross national product (GNP) for the years
1991-1996. A. B.
y' y'
y (%) 10 10
10
t t
8 1991 1996 1991 1996
6 -10 -10
4 C. D.
y' y'
2
10 10
t
0 t t
1991 1992 1993 1994 1995 1996
Year 1991 1996 1991 1996
-10 -10
8. Let C(x) be the cost (in dollars) of manufacturing x bicycles per day in a certain factory. If C(60) = 9000 and
C′(60) = 85, estimate the cost of manufacturing 61 bicycles per day.
The cost of manufacturing 61 bicycles per day would be approximately $ .
9. Consider the cost function C(x) = 6x2 + 20x + 17 (thousand dollars).
a) What is the marginal cost at production level x = 3?
b) Use the marginal cost at x = 3 to estimate the cost of producing 3.50 units.
2
c) Let R(x) = − x + 56x + 53 denote the revenue in thousands of dollars generated from the production of x units.
What is the break-even point? (Recall that the break-even point is when the revenue is equal to the cost.)
d) Compute and compare the marginal revenue and marginal cost at the break-even point. Should the company
increase production beyond the break-even point?
a) The marginal cost at production level x = 3 is $ .
b) The cost of producing 3.50 units is $ .
c) The break-even point is when units are produced. (Type an integer or a fraction.)
d) The company (1) increase production past the break even point.
(1) should
should not
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