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Magazine Chapter 3: Problem 12
Find an equation of the tangent line to the graph of \(y=g(x)\) at \(x=5\) if \(g(5)=-3\) and \(g^{\prime}(5)=4\)
Short Answer
Expert verified
The equation of the tangent line is \(y = 4x - 23\).
Step by step solution
01
Understand the Slope of the Tangent Line
The slope of the tangent line at a point on a curve is given by the derivative of the function at that point. In this case, we're given that the derivative at the point where \(x = 5\) is \(g'(5) = 4\). Therefore, the slope \(m\) of the tangent line is 4.
02
Identify the Point on the Graph
The point on the graph where the tangent line touches is at \(x = 5\). We are given \(g(5) = -3\), so the point of tangency is \((5, -3)\).
03
Substitute into the Point-Slope Form of a Line
The point-slope form of a line is given by \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope. Substitute \((x_1, y_1) = (5, -3)\) and \(m = 4\) into the equation: \[y + 3 = 4(x - 5)\].
04
Simplify the Equation
Distribute the 4 on the right side of the equation: \[y + 3 = 4x - 20\]. Subtract 3 from both sides to isolate \(y\): \[y = 4x - 23\]. This is the equation of the tangent line.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Derivative
The derivative is a fundamental concept in calculus. It represents the rate at which a function is changing at any given point. Basically, it provides the slope of the tangent line to the curve of the function at a specific point. By understanding derivatives, you can uncover trends and fluctuations in functions.
For instance, in our exercise, the derivative at the point where \(x = 5\) is given as \(g'(5) = 4\). This means that at \(x = 5\), the function is increasing at a rate of 4 units per unit change in x.
For instance, in our exercise, the derivative at the point where \(x = 5\) is given as \(g'(5) = 4\). This means that at \(x = 5\), the function is increasing at a rate of 4 units per unit change in x.
- To calculate a derivative, you apply differentiation rules, such as the power rule or chain rule.
- Derivatives can tell you where a function is increasing or decreasing, and help to find minima and maxima of functions.
- If the derivative is zero, it indicates a horizontal tangent line, which could be at a peak, a trough, or a point of inflection on the curve.
Point-Slope Form
The point-slope form is a simple way to express the equation of a line given a point and the slope. It's incredibly useful when you have precise data to input, such as in our problem. This form of the equation is given by \(y - y_1 = m(x - x_1)\), where \(m\) represents the slope and \( (x_1, y_1) \) symbolizes a point on the line.
For our tangent line equation, we used the point \( (5, -3) \) where the line touches the graph and the slope \( m = 4 \) we've calculated from the derivative.
For our tangent line equation, we used the point \( (5, -3) \) where the line touches the graph and the slope \( m = 4 \) we've calculated from the derivative.
- The point-slope form is very straightforward and efficient, especially for writing equations quickly without much rearranging.
- It's particularly useful in contexts where you need to find the line equation without additional given points.
- You can easily convert this form to other forms of the linear equation, such as slope-intercept form (\(y = mx + b\)).
Slope of Tangent Line
Understanding the slope of a tangent line is central to grasping how derivatives work in practical scenarios. In essence, the slope of the tangent line at a specific point indicates how steep the curve is at that point. The derivative gives this slope, which describes how y changes with respect to x right at that position.
For instance, a slope of 4 where \(x = 5\) means that for every unit increase in x, y increases by 4 units. This forms the backbone of finding tangent lines at any point on a function's graph.
For instance, a slope of 4 where \(x = 5\) means that for every unit increase in x, y increases by 4 units. This forms the backbone of finding tangent lines at any point on a function's graph.
- The slope of the tangent line is pivotal for visualizing how a function's output is sensitive to changes in input.
- In calculus, slopes of tangent lines are a key part of understanding motion and rates of change.
- Knowing the slope at different points can help predict future behavior and understand the function's nature better.
Function Graph
The function graph is a visual representation of how a function behaves across its domain. It plots input values (x-axis) against output values (y-axis), portraying the relationship between them visually. A function graph offers invaluable insights into the equation by showing the trends, shape, and behavior of the function at different points.
In our task involving a tangent line, the function graph informs us where the line lies, touching exactly one point, and how the slope modifies with function's changes around that point.
In our task involving a tangent line, the function graph informs us where the line lies, touching exactly one point, and how the slope modifies with function's changes around that point.
- Function graphs allow easy navigation between ideas of limits, continuity, and differentiability.
- They help identify intercepts, asymptotes, and critical points which are essential in calculus.
- Identifying various characteristics like concavity and convexity allows understanding of the function's increasing and decreasing behaviors.
