Section 1.2

12.

(a) C(d) = 0.25 d + 260 (b) C(1500) = 635. The cost would be $635. (c) The slope represents the additional cost to drive each additional mile. (d) The y-intercept represents the fixed costs of owning the car, such as insurance, parking, registration, and so on. (e) A linear model is appropriate because the cost of driving the car each additional mile is fairly constant, assuming that fuel economy and the costs of fuel and oil don't change very much.

 

Section 1.3

2.	(a) Stretch vertically by a factor of 5. (b) Shift 5 units to the right. (c) Reflect across the x-axis. (d) Reflect across the x-axis and stretch vertically by a factor of 5. (e) Compress horizontally by a factor of 5. (f) Stretch vertically by a factor of 5 and shift downward 3 units.

 

Section 2.2

2.	We can make f(x) arbitrarily close to 3 by taking x sufficintly close to 1 and less than 1; we can make f(x) arbitrarily close to 7 by taking x sufficiently close to 1 and greater than 1. The limit of f(x) as x approaches 1 does not exist in this situation, because no matter how close to 1 we move x, the function f(x) will take on values close to 3 and close to 7. So f(x) cannot be made to stay close to a single number by taking x close to 1.
4.	(a) 3 (b) 4 (c) 2 (d) Does not exist; left- and right-hand limits are different (e) 3
8.	(a) 0 (b) (infinity) (c) -(infinity) (d) -(infinity) (e) x=-5; x=0; x=4

 

Section 2.3

10.

(a) The expression on the left is undefined when x is equal to 2. The expression on the right is defined (and equal to 5) when x is equal to 2. Since the two expressions don't always give the same result, it's incorrect to assert that they are equal. (b) Since the two given expressions agree everywhere except at 2 (factor and cancel to see this), they must have the same limit as x approaches 2. This is because the limit of a function f(x) as x approaches 2 depends only on the values of f for x near 2, and does not depend on the actual value of f(2).

 

Section 2.5

6.	Here's one simple solution.
16.	There is an infinite discontinuity at x=1. The limit of f(x) as x approaches 1 from the left is -(infinity); the limit of f(x) as x approaches 1 from the right is (infinity).
20.	There is a jump discontinuity at x = 2. The limit of f(x) as x approaches 2 from the left is -1, and the limit of f(x) as x approaches 2 from the right is 0.
34.	The function has jump discontinuities at x=-1 and x=1. It is continuous from the left at -1 and continuous from the right at 1.

 

Section 2.6

8.

The slope is equal to limh -> 0(1/sqrt(1+h)-1)/h = limh -> 0(1-sqrt(1+h))/(h sqrt(1+h)) = limh -> 0((1 - (1+h))/(h sqrt(1+h) (1 + sqrt(1+h))) = limh -> 0-h/(h sqrt(1+h) (1 + sqrt(1+h))) = limh -> 0-1/(sqrt(1+h) (1+sqrt(1+h))) = -1/(sqrt(1) (1 + sqrt(1)) = -1/2. Using the point-slope form, the equation of the line is y - 1 = (-1/2)(x - 1).

 

Section 3.2

4.

(a) II. The slope is negative, then positive, then negative, and has two zeros. (b) IV. The slope is a positive constant, then a negative constant, then a positive constant. (c) I. The slope starts close to zero, then is increasingly negative, then swings through to a positive number, and then decreases toward zero. (d) III. There are three points where the tangent slope is zero.

 

Section 3.4

8.	(a) Maximum height is 180 feet. (b) On the way up, 16 ft/s; on the way down, -16 ft/s.
16.	(a) 218.75 gallons/minute (b) 187.5 gallons/minute (c) 125 gallons/minute (d) 0 gallons/minute The outflow is fastest at t=0, and slows at a constant rate to zero gallons per minute at t=40.

 

The proof

Because HTML doesn't handle mathematical notation well, I've supplied the proof separately as a PDF document. If you have the Adobe Acrobat Reader, click here to see the proof.

Section 3.6

44.	The equation of the line is y - 1 = (-sqrt(3)/2)(x - pi/6).

 

Section 3.9

14.	The boat approaches the dock at sqrt(65)/8 meters per second.

 

Section 3.10

8.	L(x) = -2 + (1/12)(x+8)
42.	dV = 0.625*pi cubic meters, or about 1,963 liters.

 

Section 4.7

2.	The numbers are 50 and -50.
12.	The minimum cost is about $163.54. The exact value is 20*(9/2)^(2/3)+180/(9/2)^(1/3).
22.	The dimensions of the largest rectangle are Width: 2*(8/3)^(1/2) Height: 16/3.
30.	The dimensions that give the largest printed area are Width = (120)^(1/2) Height = (3/2)*(120)^(1/2).
34.	The length of the shortest ladder is ((256^(1/3)+4)^2 + (64/(256^(2/3)))(256^(1/3)+4)^2)^(1/2) feet. This is about 16.65 feet.

 

Section 7.2*

2.	(1/2) ln (a) + (1/2)ln(b2 + c2)
6.	ln(6)
48.	One root is approximately 1.05800640109; the other is approximately -1.96463559749.

 

Section 7.3*

12.	The equation can be rearranged to read (ex - 3) (ex - 4) = 0, so the roots are x = ln(3) and x = ln(4).