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MATH 180: ELEMENTS OF CALCULUS I Problem Set #2

W. Stalin Rios

Due Friday, June 13, 2014 at the start of class

Limits and Continuity

One-side Limits and Limits at Infinity

1. Find the indicated limits, if they exist.

(a) lim x!�2�

f(x), lim x!�2+

f(x), lim x!�2

f(x), lim x!1

f(x), lim x!�1

f(x), if f(x) = x

2 � 2x � 3 x

2 + 5x + 6

(b) lim x!0.5�

g(x), lim x!0.5+

g(x), lim x!0.5

g(x), lim x!1

g(x), lim x!�1

g(x), if g(x) = 3x2 + 2x + 4

2x2 � 3x + 1

h(x), lim x!1+

h(x), lim x!1

h(x), lim x!�1

h(x), if h(x) = 3x3 + 5x2 � 4x � 4

2 + x � 2

(d) lim x!4�

r(x), lim x!4+

r(x), lim x!4

r(x), lim x!1

r(x), lim x!�1

r(x), if r(x) =

p x � 2 x � 4

(e) lim x!�1�

s(x), lim x!�1+

s(x), lim x!�1

s(x), lim x!1

s(x), lim x!�1

s(x), if s(x) = x

2 � 1 x

3 + 1

Example of Concentration of a Drug in the Bloodstream

2. The concentration of a certain drug, that was administered intravenously, in a patient’s bloodstream t hours after injection is given by

C(t) = C0e �kt (1)

where k is the decay constant, and is a property of the particular drug being used; C0 represents the concentration just after the first dose is administered [1]. Suppose a patient is injected a particular initial concentration C0 = 1.5 mg/ml (milligrams per milliliter) of a drug that has k = 0.4479 as the decay constant.

(a) Complete the table by computing C(t) at the given values of t.

t 0.5 1.0 2.0 5.0 10.0 20.0 C(t)

(b) Sketch the graph of C(t).

C(t) and interpret your result.

Continuous Functions

3. For all the functions in problem 1, find the values of x for which each function is continuous. Make use to use correct interval notation. Also, determine all the values of x at which each function is discontinuous.

4. Let

f(x) =

8 <

2 � 9 x + 3

if x 6= �3 kx if x = �3

(2)

Find the value of k that will make f(x) continuous on (�1, 1).

5. Let

f(x) =

⇢ x

2 � 1 if � 2  x < 1 x

3 if 1  x  2 (3)

Find lim x!1�

f(x), lim x!1+

f(x), and lim x!1

f(x). Is f(x) continuous at x = 1? Explain.

Example of Learning Curves

6. The graph of Figure 1 describes the progress a particular student made in solving a problem correctly during a chemistry exam. Here, y denotes the percent of work completed, and x is measured in minutes.

36 70 90 120

100

y(%)

Figure 1: Graph of a student’s learning curve for problem 6.

(a) Give a detailed interpretation of the graph.

(b) Using interval notation, find the values of x for which the function is continuous.

Example of Commissions

7. The base monthly salary of a salesman working on commission is $20,000. For each $50,000 of sales beyond $150,000, he is paid a $2000 commission. Sketch a graph showing his earnings as a function of the level of his sales x. Determine the values of x for which the function f is discontinuous. Make sure you use correct interval notation.

The Derivative and Di↵erentiation

For the rest of this problem set we will use the following functions:

f1(x) = 3x + 5 f2(x) = 10 � 5x f3(x) = 5x2 f4(x) = �12x 2

f5(x) = �2x2 + 3x f6(x) = 2x � x2 f7(x) = 2

3x f8(x) = x

2 � 4x + 4

f9(x) = 1

x � 2 f10(x) =

3 + 2x2 + x � 1 x

f11(x) = x 3/4 +

p x

5 f12(x) = 3x

3 � 2x3/2 + 5

f13(x) = 2

3 +

4 p x

f14(x) = � 1

4 (x�3 � x5) f15(x)

2 � 9 x + 1

f16(x) = 0.03x 3 � 0.4x2 + 0.1x

f17(x) = x +

p 2x

3x � 2 f18(x) =

(x + 7)(x2 + 7)

x � 3 f19(x) =

3 + 3 p 5x

f20(x) = (4x 2 � 3)

✓ x

2 + 2

◆

f21(x) = (x 3 + 1)5 f22(x) =

p 3x2 � 3x + 5 f23(x) =

p x

2 � 4 p x

2 + 4 f24(x) = (3x � 2)3(2x2 + 1)�2

f25(x) = x 3(4 � x)6 f26(x) = (x�2 � x�3)�4 f27(x) =

✓ x � 3 x + 3

◆7 f28(x) =

p 3x + 5

(x2 � 1)4

Slope of a Tangent Line, Average and Instantaneous Rates of Change

8. Sketch the graph of each function f1(x) through f10(x).

9. Find the slope of the secant line and the slope of the tangent line to the graph of each function f1(x) through f16(x) at any point.

10. Find the point on the graph of each function f1(x) through f16(x) where the tangent line to the curve is horizontal. Note that not all functions might have such point. Sketch these horizontal tangent lines to the curve on the same graphs from problem 8.

11. Find the average rate of change of each y i

with respect to x in the indicated interval. Here, y1 = f1(x), y2 = f2(x), and so on.

For y1:

i) from x = 1 to x = 2 ii) from x = 1 to x = 1.5 iii) from x = 1 to x = 1.1.

For y2:

i) from x = �1 to x = 0 ii) from x = �1 to x = �0.5 iii) from x = �1 to x = �0.9

For y3:

i) from x = �2 to x = �1 ii) from x = �2 to x = �1.5 iii) from x = �2 to x = �1.9

For y4:

i) from x = 3 to x = 4 ii) from x = 3 to x = 3.5 iii) from x = 3 to x = 3.1

For y5: i) from x = �1 to x = 0 ii) from x = �1 to x = �0.5 iii) from x = �1 to x = �0.9

For y6: i) from x = 2 to x = 3 ii) from x = 2 to x = 2.5 iii) from x = 2 to x = 2.1

For y7: i) from x = �4 to x = �3 ii) from x = �4 to x = �3.5 iii) from x = �4 to x = �3.9

For y8: i) from x = 3 to x = 4 ii) from x = 3 to x = 3.5 iii) from x = 3 to x = 3.1

For y9: i) from x = �3 to x = �2 ii) from x = �3 to x = �2.5 iii) from x = �3 to x = �2.9

For y10: i) from x = 1 to x = 2 ii) from x = 1 to x = 1.5 iii) from x = 1 to x = 1.1.

For y11: i) from x = 1 to x = 2 ii) from x = 1 to x = 1.5 iii) from x = 1 to x = 1.1.

For y12: i) from x = 1 to x = 2 ii) from x = 1 to x = 1.5 iii) from x = 1 to x = 1.1.

For y13: i) from x = 1 to x = 2 ii) from x = 1 to x = 1.5 iii) from x = 1 to x = 1.1.

For y14: i) from x = �1 to x = 0 ii) from x = �1 to x = �0.5 iii) from x = �1 to x = �0.9

For y15: i) from x = �2 to x = �1 ii) from x = �2 to x = �1.5 iii) from x = �2 to x = �1.9

For y16: i) from x = 2 to x = 3 ii) from x = 2 to x = 2.5 iii) from x = 2 to x = 2.1

12. Find the instantaneous rate of change of each y i

with respect to the indicated x.

For y1 : x = 1 For y2 : x = �1 For y3 : x = �2 For y4 : x = 3

For y5 : x = �1 For y6 : x = 2 For y7 : x = �4 For y8 : x = 3

For y9 : x = �3 For y10 : x = 1 For y11 : x = 1 For y12 : x = 1

For y13 : x = 1 For y14 : x = �1 For y15 : x = �2 For y16 : x = 2

13. Compare the results obtained for each function in problem 11 with that of problem 12.

14. Find the equation of the tangent line to the curve of each function f1(x) through f16(x) at the points given by the x values indicated in problem 12 .

15. The velocity of car A and car B, starting out side by side and traveling along a straight road, is given by v = f(t) and v = g(t), respectively, where v is measured in meters/second and t is measured in seconds. See Figure 2.

1 2 3 4 5

v = f(t)

v = g(t)

Figure 2: Velocities of cars A (blue) and B (red) for problem 15.

(a) What can you say about the velocity and acceleration of the two cars at t = 2 seconds?

(b) What can you say about the velocity and acceleration of the two cars at t = 4 seconds?

Example of Revenue of a Travel Agency

16. Suppose that the total revenue realized by the Galasam Travel Agency is R = f(x) thousand dollars if x thousand dollars are spent on advertising.

(a) What does f(b) � f(a)

b � a (0 < a < b)

measure? What are the units?

(b) What does f0(x) measure? Give units.

(c) Given that f0(25) = 4, what is the approximate change in the revenue if Galasam increases its advertising budget from $25,000 to $26,000?

17. Let x and f(x) represent the given quantities. Fix x = a and let h be a small positive number. Give an interpretation of the quantities

f(a + h) � f(a) h

and lim h!0

f(a + h) � f(a) h

(a) x denotes time and f(x) denotes the population of giant turtles at time x.

(b) x denotes time and f(x) denotes a country’s industrial production.

Rules of Di↵erentiation

18. Find the derivative of each function f1(x) through f28(x) by using the rules of di↵erentia- tion.

19. Suppose f(x) and g(x) are functions that are di↵erentiable at x = 1 and that f(1) = 3, f

0(1) = �2, g(1) = �1, and g0(1) = 2. Find the value of h0(1).

(a) h(x) = f(x)g(x)

(b) h(x) = f(x)g(x)

f(x) � g(x)

(d) h(x) = f(x 2

2 + 1

2 )

20. Suppose g(x) = (x2 � 1)f(x) and it is known that f(3) = 2 and f0(3) = �2. Evaluate g

0(3).

References

[1] Geometric series and e↵ective medicine dosage. Mathematical Science, Worcester Polytech- nic Institute.