Problem 3

Question

For $f(x)=3 x$ and $g(x)=1+3 x$, find $f(4+h)$ and $g(4+h)$ and $f^{\prime}(4)$ and $g^{\prime}(4)$. Sketch the graphs of $f$ and $g-$ why do they have the same slope?

Step-by-Step Solution

Verified

Answer

The functions have slope 3, hence parallel graphs.

1Step 1: Evaluate f(4+h)

Start by substituting $4+h$ into the function $f(x) = 3x$. This gives us: \[f(4+h) = 3(4+h) = 3 \cdot 4 + 3 \cdot h = 12 + 3h.\]

2Step 2: Evaluate g(4+h)

Substitute $4+h$ into the function $g(x) = 1 + 3x$. This gives: \[g(4+h) = 1 + 3(4+h) = 1 + 3 \cdot 4 + 3 \cdot h = 1 + 12 + 3h = 13 + 3h.\]

3Step 3: Find f'(x) and evaluate at x=4

The function $f(x) = 3x$ is a linear function. Its derivative is: $f'(x) = 3$. Therefore, $f'(4) = 3$.

4Step 4: Find g'(x) and evaluate at x=4

The function $g(x) = 1 + 3x$ is also linear. Its derivative is: $g'(x) = 3$. Hence, $g'(4) = 3$.

5Step 5: Graphing Explanation and Slope Comparison

Both $f(x)$ and $g(x)$ are linear functions with the form $y = mx + b$, where the slope $m$ is 3 for both. This explains why the graphs are parallel (same slope). They have different y-intercepts, 0 for $f(x)$ and 1 for $g(x)$, but share the same slope (3), explaining the same gradient.

Key Concepts

Linear functionsSlope of a lineGraphing functions

Linear functions

Linear functions are among the simplest types of functions in mathematics. They are defined as functions that graph to a straight line. A linear function has the form $ y = mx + b $, where:

$ m $ represents the slope of the line.
$ b $ represents the y-intercept, which is the point where the line crosses the y-axis.

For the given functions $ f(x) = 3x $ and $ g(x) = 1 + 3x $, both are linear due to their straight-line form. Each function has a slope of 3, which means for every 1 unit increase in $ x $, $ y $ increases by 3 units. The difference between $ f(x) $ and $ g(x) $ is the y-intercept: 0 for $ f(x) $ and 1 for $ g(x) $. This only shifts the line up or down on the graph but does not change its slope. Understanding linear functions is crucial because they model many real-world situations where a constant rate of change is present.

Slope of a line

The slope of a line is a measure of its steepness and direction. It is defined as the "rise over run," meaning how much the line goes up or down (rise) for each unit it moves left or right (run). In the slope-intercept form $ y = mx + b $, the coefficient $ m $ before $ x $ is the slope.

If $ m $ is positive, the line rises from left to right. For example, both functions $ f(x) = 3x $ and $ g(x) = 1 + 3x $ have slopes of 3, indicating they rise steeply.
When the slope is zero, the line is flat, meaning it has no rise.
A negative slope means the line falls from left to right.

With $ m = 3 $, both our example functions have a "rise" of 3 units up for each "run" of 1 unit to the right, demonstrating why their slopes cause parallel lines when graphed. The understanding of a slope helps in interpreting linear functions and predicting how changes in the variable $ x $ affect the outcome.

Graphing functions

Graphing functions is a fundamental skill in mathematics that involves plotting the points of a function on a set of axes, usually the x and y axes. For linear functions like $ f(x) = 3x $ and $ g(x) = 1 + 3x $, this process involves:

Identifying the slope and intercept from the equation.
Marking the y-intercept on the graph.
Using the slope to determine the direction and steepness of the line.

Both functions have the same slope, so their graphs will be parallel. However, they have different y-intercepts. The graph of $ f(x) = 3x $ will cross the y-axis at 0, while $ g(x) = 1 + 3x $ will cross at 1. Because they are parallel, with the same slope but different intercepts, it illustrates that linear functions' lines can be shifted vertically without altering their orientation. Understanding graphing helps visually demonstrate and analyze how a function behaves across different values of $ x $. It bridges the gap between algebraic expressions and their geometric representation.

Problem 3

Problem 4

Other exercises in this chapter

Problem 3

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Problem 3

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Problem 4

Decide whether $A \Rightarrow B$ or $B \Rightarrow A$ or neither or both: (a) \(A=\left[a_{n} \rightarrow 1\right] \quad B=\left[-a_{n} \rightarrow-1\right]

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Problem 4

Find the derivatives of the functions in exercise. Even if $n$ is negative or a fraction, the derivative of $x^{n}$ is $n x^{n-1}$. $$ 1+(7 / x)+\left(5 /

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