Mat 220 Calculus I Chapter 1 Exam Name_______________________________

Spring, 2008 Key

Directions: Show all your algebraic steps to justify your solutions. Please signify your
solutions clearly.

#1. Find a piecewise function to define the graph below. (3 pts)

1.5

0.5

-2.5 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5

-0.5

-1

-1.5

 − x − 1 if x ≤ −1

f ( x ) =  1 − x 2 if −1 < x ≤ 1
 − x + 1 if x >1

**We did one similar in class.
#2. Find the domain for the following function. (3 pts)
x −2
f ( x) =
4x +1 − 5
x ⇒ x≥0
−1
4x +1 ⇒ 4x + 1 ≥ 0 → x ≥
4
4 x + 1 − 5 ≠ 0 → 4 x + 1 ≠ 25 → x ≠ 6
D f : { x | x ≥ 0, x ≠ 6} or x ∈ [ 0, 6 ) ∪ ( 6, ∞ )
**We did one similar in class. Remember, the domain violators?

x
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-2
#3. Suppose you have a function f(x). Write in the blanks the appropriate expression
for the stated transformation. (example:a vertical shift of 5 units up ⇒ f(x) + 5 ).
(2pts).

a. Shift 5 units downward f ( x) − 5

b. Shift 5 units to the right f ( x − 5)
c. Reflect across the y-axis f (−x)
d. Reflect across the x-axis − f ( x)
e. Vertical stretch by a factor of 5 5 f ( x)
x
f. Horizontal stretch by a factor of 5 f 
5
f ( x)
g. Vertical compression by a factor of 5
5
h. Horizontal compression by a factor of 5 f (5x )

** See practice test and chapter review.

#4. Let f(x) = ln(x), g(x) = x2 – 9, and h(x) = 1 + x

x-5

A. Find f(g(x)) and state its domain. (2 pts)

f ( g ( x ) ) = ln ( x 2 − 9 ) ;
x2 − 9 > 0 ⇔ x2 > 9 ⇔ x > 3
so,
D f ( g ( x ) ) : { x | x > 3} or { x | x < −3 or x > 3} or x ∈ ( −∞, −3) ∪ ( 3, ∞ )
B. Find ( h  f )( x ) and state its domain. (2 pts)
1 + ln ( x )
( h  f )( x ) = ;
ln ( x ) − 5
ln ( x ) ⇒ x > 0
ln ( x ) − 5 ≠ 0 ⇔ ln ( x ) ≠ 5 ⇔ x ≠ e5
∴ D( h f )( x ) : { x | x > 0, x ≠ e5 } or x ∈ ( 0, e5 ) ∪ ( e5 , ∞ )

** We did many domain and composition examples.

#5. Find the inverse of the following function. (3 pts)

2x + 1
f ( x) =
1 − 3x
2x + 1 2 y +1
y= →x=
1 − 3x 1− 3y
x (1 − 3 y ) = 2 y + 1
x − 3 xy = 2 y + 1
x − 1 = 2 y + 3 xy
x − 1 = ( 2 + 3x ) y
x −1 x −1
y= → f −1 ( x ) =
3x + 2 3x + 2

** See practice test and the example from the class notes.

#6. The half-life of plutonium246 is 13 years. (4 pts)

a. How much of an initial 5 gram sample remains after 10 years?

t
 1  half −life
Q ( t ) = Q0  
2
10
 1 13
Q (10 ) = 5   ≈ 2.96 g
2
b. Find the time required for the 5 gram sample to decay to 1 gram.

t
 1 13 t ln ( 0.2 )
1 = 5   ⇔ 0.2 = ( 0.5 )13 ⇔ t = 13
2 ln ( 0.5 )
t ≈ 30.19 years

** I did number 26 from section 1-5 of the text. There was also one assigned in the
review page.
#7.
A. List the four ways to represent a function. (2 points)
i. In words (verbally).
ii. With a formula.
iii. With a table of values.
iv. With a graph.

B. List the three domain violators. (1.5 points)

i. Division by zero.
ii. Negative radicands in even indexed
radicals.
iii. A context conflict.

** We discussed these in class and pages 14 & 17 also elaborate on this.

#8. Use the given graphs to evaluate each expression or explain why it is undefined.
(3.5 points)

( f  g )( 2 ) = f ( g ( 2 ) ) = f ( 0 ) = 1
( g  f )( 2 ) = g ( f ( 2 ) ) = g ( 3) = 1
( f  f )( 2 ) = f ( f ( 2 ) ) = f ( 3) = 4
( g  g )( 2 ) = g ( g ( 2 ) ) = g ( 0 ) = 4
( f + g )( 2 ) = f ( 2 ) + g ( 2 ) = 3 + 0 = 3
 f  f ( 2) 3
  ( 2) = = → undefined /* division by zero
g g ( 2) 0
g −1 ( 2 ) → Does not exist /* Fails the Horizontal Line Test.
#9. Is the following function odd, even, or neither? Back up your answer algebraically as
shown in class and the homework. (3 pts)
1
f ( x ) = x5 − +x
x3
5 1 1
f (−x) = (−x) − 3
+ ( − x ) = − x5 + − x ≠ f ( x ) ∴ not even
(−x) x3
 1  1
− f ( − x ) = −  − x5 + 3 − x  = x5 − 3 + x = f ( x ) ∴ f ( x ) is odd .
 x  x

** If you missed this it is probably because you used your calculator!

#10. Observe the sphere. Its volume and surface areas are given as: (2 pts)

4π r 3
Vsphere = and SAsphere = 4π r 2
3

Vsphere
A. Write the ratio of the volume to the surface area, , in reduced
SAsphere
form.
4π r 3
Vsphere = and SAsphere = 4π r 2
3
4π r 3
Vsphere 4π r 3 r
= 32 = 2
= /* this simplifies nicely!
SAsphere 4π r 3 ⋅ 4π r 3

B. Using part A, what is the radius of a sphere whose volume is 27 cubic feet
and whose surface area is 81 square feet?

From A
Vsphere r 27 ft 3 r 3 ⋅ 27 ft 3
= ⇔ = ⇔r= = 1 ft
SAsphere 3 81 ft 2 3 81 ft 2

** Every year students make this so much harder than it needs to be. Part of the reason
is that they did not take problems 47 – 52 of section 1-1 seriously.

1
( )
#11. If g ( x ) = cos ln ( x ) + e
x −1
+
x
−1
, find g ( 3 ) . (2 pts)

1
We note that : g (1) = cos ( ln (1) ) + e1−1 + = cos ( 0 ) + e0 + 1 = 3
1
−1
∴ g ( 3) = 1

** I specifically did two problems in class like this.

#12Bonus*. Show algebraically that the function (3 pts)

f ( x) = ln( x + x 2 + 1) is odd.
** This problem is much more difficult that it appears.

 x2 + 1 + x 
−1  1  
= ln  
1  ( ) 
− f (− x) = − ln(− x + (− x) 2 + 1) = ln ( x2 + 1 − x ) = ln    
2
 x +1 − x 
2
  x +1 − x 
 ( )
x2 + 1 + x 

 x2 + 1 + x
(
= ln  2
)  = ln  x + x2 + 1 
 x +1− x
2 


 1 

(
 = ln x + x 2 + 1 = f ( x) )
 
∴ f ( x) is odd.

If your test grade is not as high as you would like ask yourself these questions.
How many hours did I spend studying several days before the exam?
Have I devoted enough time to the homework problems and am I really taking good notes
from the lectures?
How many times have I reread my notes, perhaps even recopying them?
Am I using proper mathematical notation in my answers or am I making up my own
symbolism?
Is there a clear flow to my work or is stuff written haphazardly?
Did I answer all parts of the question and include units when needed?
If I were the teacher grading this paper, would I be able to follow the algebraic
justifications of the solutions?

I have very high standards that I hold all of my Calculus students to. I am preparing you
for a multitude of universities, colleges, and careers. You will need to devote an
unusually high amount of time to excel in this class.