Question
Select the correct answer.
Consider the functions f and g in the tables below.
f(x) = 90x2 + 180x + 92
x y
0 92
1 362
2 812
3 1,442
4 2,252
5 3,242
g(x) = 6x
x y
0 1
1 6
2 36
3 216
4 1,296
5 7,776
Which of the following statements is true?
A.
For every value of x, the rate of change of g exceeds the rate of change of f.
B.
At approximately x = 4.39, the rate of change of f is equal to the rate of change of g.
C.
As x increases, the rate of change of f exceeds the rate of change of g.
D.
As x increases, the rate of change of g exceeds the rate of change of f.
Consider the functions f and g in the tables below.
f(x) = 90x2 + 180x + 92
x y
0 92
1 362
2 812
3 1,442
4 2,252
5 3,242
g(x) = 6x
x y
0 1
1 6
2 36
3 216
4 1,296
5 7,776
Which of the following statements is true?
A.
For every value of x, the rate of change of g exceeds the rate of change of f.
B.
At approximately x = 4.39, the rate of change of f is equal to the rate of change of g.
C.
As x increases, the rate of change of f exceeds the rate of change of g.
D.
As x increases, the rate of change of g exceeds the rate of change of f.
Answers
GPT-4o mini
To analyze the two functions \( f(x) = 90x^2 + 180x + 92 \) and \( g(x) = 6x \), we first need to understand their rates of change.
1. **Finding the derivatives to assess the rates of change**:
- The derivative of \( f(x) \) with respect to \( x \) is:
\[
f'(x) = 180x + 180
\]
- The derivative of \( g(x) \) with respect to \( x \) is:
\[
g'(x) = 6
\]
2. **Comparison of rates of change**:
- \( f'(x) = 180x + 180 \) is a linear function that increases as \( x \) increases. This means the rate of change of \( f \) increases significantly as \( x \) gets larger.
- \( g'(x) \), being a constant \( 6 \), does not change as \( x \) changes.
3. **At specific values**:
- At \( x = 0 \):
\[
f'(0) = 180(0) + 180 = 180, \quad g'(0) = 6 \quad \Rightarrow f'(0) > g'(0)
\]
- At \( x = 1 \):
\[
f'(1) = 180(1) + 180 = 360, \quad g'(1) = 6 \quad \Rightarrow f'(1) > g'(1)
\]
- At \( x = 2 \):
\[
f'(2) = 180(2) + 180 = 540, \quad g'(2) = 6 \quad \Rightarrow f'(2) > g'(2)
\]
- This trend continues since the derivative of \( f(x) \) is positive and increasing for all \( x \).
4. **Rising rate of change of f**:
- As \( x \) increases beyond 0, \( f'(x) \) will always exceed \( g'(x) \) because the value of \( f'(x) \) increases with \( x \) and \( g'(x) \) remains a constant value of \( 6 \).
5. **Evaluating statements**:
- **A**: Incorrect. The rate of change of \( g \) cannot exceed \( f \); \( f' > g' \) for all \( x \).
- **B**: Possible but needs checking for the approximate equality. However, observations indicate \( f' \) exceeds \( g' \) for all \( x \).
- **C**: True. As shown, \( f' > g'\) for all values of \( x \).
- **D**: Incorrect. \( g' \) does not exceed \( f' \) for any \( x \).
Therefore, the correct answer is:
**C. As x increases, the rate of change of f exceeds the rate of change of g.**
1. **Finding the derivatives to assess the rates of change**:
- The derivative of \( f(x) \) with respect to \( x \) is:
\[
f'(x) = 180x + 180
\]
- The derivative of \( g(x) \) with respect to \( x \) is:
\[
g'(x) = 6
\]
2. **Comparison of rates of change**:
- \( f'(x) = 180x + 180 \) is a linear function that increases as \( x \) increases. This means the rate of change of \( f \) increases significantly as \( x \) gets larger.
- \( g'(x) \), being a constant \( 6 \), does not change as \( x \) changes.
3. **At specific values**:
- At \( x = 0 \):
\[
f'(0) = 180(0) + 180 = 180, \quad g'(0) = 6 \quad \Rightarrow f'(0) > g'(0)
\]
- At \( x = 1 \):
\[
f'(1) = 180(1) + 180 = 360, \quad g'(1) = 6 \quad \Rightarrow f'(1) > g'(1)
\]
- At \( x = 2 \):
\[
f'(2) = 180(2) + 180 = 540, \quad g'(2) = 6 \quad \Rightarrow f'(2) > g'(2)
\]
- This trend continues since the derivative of \( f(x) \) is positive and increasing for all \( x \).
4. **Rising rate of change of f**:
- As \( x \) increases beyond 0, \( f'(x) \) will always exceed \( g'(x) \) because the value of \( f'(x) \) increases with \( x \) and \( g'(x) \) remains a constant value of \( 6 \).
5. **Evaluating statements**:
- **A**: Incorrect. The rate of change of \( g \) cannot exceed \( f \); \( f' > g' \) for all \( x \).
- **B**: Possible but needs checking for the approximate equality. However, observations indicate \( f' \) exceeds \( g' \) for all \( x \).
- **C**: True. As shown, \( f' > g'\) for all values of \( x \).
- **D**: Incorrect. \( g' \) does not exceed \( f' \) for any \( x \).
Therefore, the correct answer is:
**C. As x increases, the rate of change of f exceeds the rate of change of g.**