Problem 5
Question
Find the slope of the tangent line to the graph of \(f\) at the given point. $$f(x)=4 x^{2}-3 x, \quad \text { at }(-1,7)$$
Step-by-Step Solution
Verified Answer
The slope of the tangent line at \((-1, 7)\) is \(-11\).
1Step 1: Differentiate the Function
To find the slope of the tangent line at a specific point, we first need to find the derivative of the function. The derivative, \( f'(x) \), represents the slope of the tangent line at any point \( x \) on the curve. For the function \( f(x) = 4x^2 - 3x \), we differentiate term by term. The derivative of \( 4x^2 \) is \( 8x \), and the derivative of \( -3x \) is \( -3 \). Therefore, the derivative is:\[ f'(x) = 8x - 3. \]
2Step 2: Evaluate the Derivative at the Given Point
Next, we need to find the slope of the tangent line at the specific point \( x = -1 \). This is done by substituting \( x = -1 \) into the derivative \( f'(x) \). So we compute: \( f'(-1) = 8(-1) - 3 = -8 - 3 = -11. \) This means the slope of the tangent line at \( x = -1 \) is \( -11 \).
3Step 3: Verify the Given Point on the Function
To ensure that the point \((-1, 7)\) lies on the graph of \( f(x) \), substitute \( x = -1 \) into \( f(x) \) to confirm that it equals \( 7 \). Substitute into the function: \( f(-1) = 4(-1)^2 - 3(-1) = 4 \times 1 + 3 = 7 \). The calculation confirms that \((-1, 7)\) is indeed a point on the curve.
Key Concepts
DerivativeFunction EvaluationGraph Analysis
Derivative
Calculating a derivative is one of the crucial steps in determining the slope of a tangent line. A derivative represents the rate of change of a function with respect to its variable.
In simple terms, it tells us how the function is behaving at any point. Finding the derivative involves differentiating the function term by term. For the function provided, \(f(x) = 4x^2 - 3x\), the derivative is calculated by applying the power rule.
This rule states that for any term \(ax^n\), its derivative is \(nax^{n-1}\). Thus:
In simple terms, it tells us how the function is behaving at any point. Finding the derivative involves differentiating the function term by term. For the function provided, \(f(x) = 4x^2 - 3x\), the derivative is calculated by applying the power rule.
This rule states that for any term \(ax^n\), its derivative is \(nax^{n-1}\). Thus:
- The derivative of \(4x^2\) is \(8x\).
- The derivative of \(-3x\) is \(-3\).
Function Evaluation
Function evaluation is an important process which involves substituting values into a function to find specific outcomes.
In this problem, after finding the derivative \(f'(x) = 8x - 3\), evaluating it at a given point helps us find the slope of the tangent line at that location.
Substituting \(x = -1\) into \(f'(x)\) gives us \(f'(-1) = 8(-1) - 3 = -11\). This value, \(-11\), represents how steep the line is at \(x = -1\). This process confirms the behavior of the function around that point, helping us understand its instantaneous rate of change.
Being able to effectively evaluate functions assists students in comprehending how functions behave at specific points, which is vital in calculus and beyond.
In this problem, after finding the derivative \(f'(x) = 8x - 3\), evaluating it at a given point helps us find the slope of the tangent line at that location.
Substituting \(x = -1\) into \(f'(x)\) gives us \(f'(-1) = 8(-1) - 3 = -11\). This value, \(-11\), represents how steep the line is at \(x = -1\). This process confirms the behavior of the function around that point, helping us understand its instantaneous rate of change.
Being able to effectively evaluate functions assists students in comprehending how functions behave at specific points, which is vital in calculus and beyond.
Graph Analysis
Graph analysis is a visual way to understand functions and their growth, slopes, or other characteristics.
Once we determine the slope using derivatives and function evaluation, examining the graph of the function provides insight into the result.
In this exercise, it's vital to verify that the point \((-1,7)\) lies on the function's graph. For the function \(f(x) = 4x^2 - 3x\), substituting \(x = -1\) in the original function gives \(f(-1) = 7\), confirming the point. Graphically, the tangent line at \((-1,7)\) is a straight line whose steepness is given by the slope \(-11\).
Graph analysis helps students visualize abstract concepts and confirm calculations, fostering a deeper understanding of how numbers and graphs relate in mathematics.
Once we determine the slope using derivatives and function evaluation, examining the graph of the function provides insight into the result.
In this exercise, it's vital to verify that the point \((-1,7)\) lies on the function's graph. For the function \(f(x) = 4x^2 - 3x\), substituting \(x = -1\) in the original function gives \(f(-1) = 7\), confirming the point. Graphically, the tangent line at \((-1,7)\) is a straight line whose steepness is given by the slope \(-11\).
Graph analysis helps students visualize abstract concepts and confirm calculations, fostering a deeper understanding of how numbers and graphs relate in mathematics.
Other exercises in this chapter
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