Exercises Topic 1

Worksheet 2 (17-18). Differentiation-STEPHANIE REINA FriOct/6/2017, 22/13




Student: _____________________ 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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