Problem 52
Question
Find a line that is tangent to the graph of the given function \(f\) and that is parallel to the line \(y=12 x\). $$ f(x)=x^{2}-4 x+2 $$
Step-by-Step Solution
Verified Answer
The tangent line is \( y = 12x - 62 \).
1Step 1: Identify the Slope of the Given Line
The line given is \( y = 12x \). The slope of this line is the coefficient of \( x \), which is 12. Hence, the slope of any line that is parallel to \( y = 12x \) is also 12.
2Step 2: Find the Derivative of the Function
The derivative \( f'(x) \) will give us the slope of the tangent line to the curve \( f(x) = x^{2} - 4x + 2 \) at any point \( x \). So, we calculate \( f'(x) = \frac{d}{dx}(x^{2} - 4x + 2) = 2x - 4 \).
3Step 3: Set the Derivative Equal to the Slope of the Given Line
We want the slope of the tangent line, given by \( f'(x) \), to be equal to 12. So, we set the equation: \( 2x - 4 = 12 \).
4Step 4: Solve for \( x \)
Solve the equation \( 2x - 4 = 12 \) to find the \( x \)-coordinate of the point where the tangent to the curve is parallel to the given line. Solving gives: \( 2x = 16 \) which simplifies to \( x = 8 \).
5Step 5: Find the Corresponding \( y \) Coordinate
Substitute \( x = 8 \) back into the original function \( f(x) \) to find the corresponding \( y \)-value. \( f(8) = 8^{2} - 4 \times 8 + 2 = 64 - 32 + 2 = 34 \). So the point is \((8, 34)\).
6Step 6: Write the Equation of the Tangent Line
Now, using the point \((8, 34)\) and slope 12, we can write the equation of the tangent line in point-slope form: \( y - 34 = 12(x - 8) \). Simplifying this gives \( y = 12x - 62 \).
Key Concepts
DerivativeSlopeParallel LinesFunction
Derivative
A derivative is a fundamental concept in calculus, which helps us understand how a function changes at any given point. Think of it as a tool to measure the instantaneous rate of change. When applied to a function, the derivative represents the slope of the tangent line at any given point on the graph of that function.
In the exercise, we are working with the function \(f(x) = x^2 - 4x + 2 \). By calculating the derivative of this function, we get \(f'(x) = 2x - 4 \). This new equation, \(2x - 4 \), tells us the slope of the tangent line for any given value of \(x \).
In the exercise, we are working with the function \(f(x) = x^2 - 4x + 2 \). By calculating the derivative of this function, we get \(f'(x) = 2x - 4 \). This new equation, \(2x - 4 \), tells us the slope of the tangent line for any given value of \(x \).
- To find where this tangent is parallel to another line, we need the slopes to match.
- Derivatives are critical in finding these parallel tangent lines because they allow us to analyze and match slopes precisely.
Slope
The slope is a measure of the steepness and direction of a line. For a straight line, it's the ratio of how much the line goes up or down for each unit it goes across.
In mathematical terms, it's represented as \(m\) in the equation \(y=mx+b\), where \(m\) is the slope. When dealing with curves, the slope can change at each point along the graph.
In this exercise, we're considering two types of slopes:
In mathematical terms, it's represented as \(m\) in the equation \(y=mx+b\), where \(m\) is the slope. When dealing with curves, the slope can change at each point along the graph.
In this exercise, we're considering two types of slopes:
- The slope of the existing line \(y=12x\), which is easy to spot because it's simply \(12\).
- The slope of the tangent to the curve, determined by the derivative \(f'(x) = 2x - 4\).
Parallel Lines
Parallel lines are lines in a plane that never meet. They have the same slope but different intercepts.
In the context of this exercise, we're searching for a tangent line to the function \(f(x)\) that is parallel to the line \(y=12x\). Because they are parallel, both lines must have the same slope of \(12\).
In the context of this exercise, we're searching for a tangent line to the function \(f(x)\) that is parallel to the line \(y=12x\). Because they are parallel, both lines must have the same slope of \(12\).
- To find a specific point on \(f(x)\) where the tangent line is parallel, we need to match the slope given by its derivative with the slope of the line \(y=12x\).
- That exact match is what makes the lines parallel, ensuring they never intersect.
Function
A function is a relation between sets that assigns to each element of a domain exactly one element of a range.
In simpler terms, a function describes how each input (or \(x\)-value) affects the output (or \(y\)-value). The function in this exercise is \(f(x) = x^2 - 4x + 2\). Knowing the derivative or slope at any point on this function helps us understand how steep the curve is at that point.
The process in the exercise:
In simpler terms, a function describes how each input (or \(x\)-value) affects the output (or \(y\)-value). The function in this exercise is \(f(x) = x^2 - 4x + 2\). Knowing the derivative or slope at any point on this function helps us understand how steep the curve is at that point.
The process in the exercise:
- Starts with finding the slope or derivative of \(f(x)\), which reveals the behavior of the function.
- This allows us to identify the point where the function's behavior (slope) aligns with another line's behavior.
Other exercises in this chapter
Problem 52
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If \(f(-x)=f(x)\) for every \(x,\) then \(f\) is said to be an even function. If \(f(-x)=-f(x),\) for every \(x,\) then \(f\) is said to be an odd function. Ass
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Differentiate the given expression with respect to \(x\). $$ \tanh ^{-1}(\cosh (x)) $$
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