Problem 3
Question
(a) If you are given an equation for the tangent line at the point \((a, f(a))\) on a curve \(y=f(x),\) how would you go about finding \(f^{\prime}(a) ?\) (b) Given that the tangent line to the graph of \(y=f(x)\) at the point (2,5) has the equation \(y=3 x-1,\) find \(f^{\prime}(2)\) (c) For the function \(y=f(x)\) in part (b), what is the instantaneous rate of change of \(y\) with respect to \(x\) at \(x=2 ?\)
Step-by-Step Solution
Verified Answer
(a) Use the slope of the tangent line to find \(f'(a)\). (b) \(f'(2) = 3\). (c) The instantaneous rate of change of \(y\) at \(x=2\) is 3.
1Step 1: Understanding the Tangent Line Equation
The equation of a tangent line at a point \((a, f(a))\) on the curve \(y=f(x)\) provides two pieces of critical information. First, the point of tangency is \((a, f(a))\). Second, the slope of the tangent line, which is given in the equation, is equal to the derivative of the function \(f(x)\) at that point, \(f'(a)\).
2Step 2: Identify the Slope from the Tangent Line Equation
Given the equation of the tangent line\(y = 3x - 1\), identify the slope \(m\). In the equation \(y = mx + b\), \(m\) represents the slope. For the given equation, the slope \(m = 3\). This is the value of \(f'(2)\).
3Step 3: Use Slope to Determine the Derivative at a Point
Since the slope of the tangent line at \((2, 5)\) is obtained from the line \(y = 3x - 1\), conclude that the derivative \(f'(a)\) is equal to 3. Therefore, \(f'(2) = 3\).
4Step 4: Understand the Instantaneous Rate of Change
The instantaneous rate of change of a function \(y=f(x)\) with respect to \(x\) at \(x=a\) is given by the derivative \(f'(a)\). As calculated from the tangent line's slope, this value is also 3 at \(x=2\).
5Step 5: Interpret the Instantaneous Rate of Change
The instantaneous rate of change of \(y\) with respect to \(x\) at \(x=2\) is the same as the derivative \(f'(2)\). Therefore, it is 3, meaning the function \(y\) increases by 3 units for every 1 unit increase in \(x\) at \(x=2\).
Key Concepts
Instantaneous Rate of ChangeTangent LineSlope of the Tangent LineDerivative at a Point
Instantaneous Rate of Change
The concept of instantaneous rate of change helps us understand how a quantity changes at a specific instant or point. When you look at a function on a graph, this rate tells you how steep the curve is at a particular point. Imagine you're driving a car and looking at the speedometer. The speedometer shows your speed at any given instant—that's an example of an instantaneous rate of change.
In calculus, we find the instantaneous rate of change by calculating the derivative of the function at a specific point. For the function \(y=f(x)\) with respect to \(x\) at \(x=a\), this is represented as \(f'(a)\). It gives you a precise slope of the curve at that point, indicating how quickly or slowly the function value \(y\) is changing with respect to \(x\)
At \(x=2\), for instance, in the given exercise, the derivative \(f'(2)\) equals 3. This means that the function is increasing by 3 units for every 1 unit of change in \(x\) at that instant.
Understanding this concept is crucial because it offers a snapshot of the behavior of functions, which is foundational in calculus and real-world applications.
In calculus, we find the instantaneous rate of change by calculating the derivative of the function at a specific point. For the function \(y=f(x)\) with respect to \(x\) at \(x=a\), this is represented as \(f'(a)\). It gives you a precise slope of the curve at that point, indicating how quickly or slowly the function value \(y\) is changing with respect to \(x\)
At \(x=2\), for instance, in the given exercise, the derivative \(f'(2)\) equals 3. This means that the function is increasing by 3 units for every 1 unit of change in \(x\) at that instant.
Understanding this concept is crucial because it offers a snapshot of the behavior of functions, which is foundational in calculus and real-world applications.
Tangent Line
A tangent line is a straight line that touches a curve at exactly one point without crossing it (in the immediate vicinity). Think of a tangent line as the best straight-line approximation of the curve at that specific point.
When we have a curve described by a function \(y=f(x)\), the tangent line at a point \(x=a\) will have the same rate of change as the function at that point. In mathematical terms, the slope of this tangent line is the derivative of the function at that point, which is \(f'(a)\).
For example, if the equation of the tangent line at \((a, f(a))\) is given, it reveals crucial insights about the behavior of the function near that point. In our exercise, the tangent line at \((2, 5)\) is described by \(y = 3x - 1\). This line not only touches the curve at the given point but also provides the slope of the tangent line as 3. This slope directly tells us the derivative \(f'(2)\).
Through tangent lines, we bridge the gap between geometry (shapes and sizes) and calculus (rates of change), offering powerful tools to assess and predict changes in various scenarios.
When we have a curve described by a function \(y=f(x)\), the tangent line at a point \(x=a\) will have the same rate of change as the function at that point. In mathematical terms, the slope of this tangent line is the derivative of the function at that point, which is \(f'(a)\).
For example, if the equation of the tangent line at \((a, f(a))\) is given, it reveals crucial insights about the behavior of the function near that point. In our exercise, the tangent line at \((2, 5)\) is described by \(y = 3x - 1\). This line not only touches the curve at the given point but also provides the slope of the tangent line as 3. This slope directly tells us the derivative \(f'(2)\).
Through tangent lines, we bridge the gap between geometry (shapes and sizes) and calculus (rates of change), offering powerful tools to assess and predict changes in various scenarios.
Slope of the Tangent Line
The slope of the tangent line is the rate at which the tangent line rises or falls as you move along the \(x\)-axis. It's a crucial concept because it represents the derivative of a function at a particular point and thus conveys the function's instantaneous rate of change.
The slope of a line can be interpreted as the "rise over run," i.e., the change in \(y\) divided by the change in \(x\). For a tangent line defined by the equation \(y = mx + b\), the slope \(m\) is key because it equals the derivative \(f'(a)\) at the point of tangency \((a, f(a))\).
In the given exercise, the tangent line's slope is found in the equation \(y = 3x - 1\), where the slope \(m = 3\). This means that as \(x\) increases by 1 unit, \(y\) increases by 3 units at \(x=2\). This crucial insight allows us to understand how sharply or gently the curve is changing at that point.
Recognizing the slope of the tangent line helps us interpret and predict the behavior of changing systems, from physics to finance.
The slope of a line can be interpreted as the "rise over run," i.e., the change in \(y\) divided by the change in \(x\). For a tangent line defined by the equation \(y = mx + b\), the slope \(m\) is key because it equals the derivative \(f'(a)\) at the point of tangency \((a, f(a))\).
In the given exercise, the tangent line's slope is found in the equation \(y = 3x - 1\), where the slope \(m = 3\). This means that as \(x\) increases by 1 unit, \(y\) increases by 3 units at \(x=2\). This crucial insight allows us to understand how sharply or gently the curve is changing at that point.
Recognizing the slope of the tangent line helps us interpret and predict the behavior of changing systems, from physics to finance.
Derivative at a Point
The derivative at a point provides a precise measurement of how a function \(y=f(x)\) is changing at a specific \(x\)-value. In essence, it tells us the slope of the tangent line to the curve at that specific point.
To find the derivative at a particular point, say \(x=a\), we calculate \(f'(a)\). This derivative holds valuable information—it is the instantaneous rate of change at that point and gives us the slope of the tangent line touching the curve at \(x=a\).
In our exercise, for instance, we're given that the derivative at \(x=2\), denoted as \(f'(2)\), equals 3. This means the tangent line at this point has a slope of 3, indicating a fairly linear increase in \(y\) as \(x\) increases around \(x=2\).
Derivatives at a point are a cornerstone of calculus. They allow us to understand the behavior of functions in a most granular and precise manner, providing essential insights into the dynamics of various real-world phenomena.
To find the derivative at a particular point, say \(x=a\), we calculate \(f'(a)\). This derivative holds valuable information—it is the instantaneous rate of change at that point and gives us the slope of the tangent line touching the curve at \(x=a\).
In our exercise, for instance, we're given that the derivative at \(x=2\), denoted as \(f'(2)\), equals 3. This means the tangent line at this point has a slope of 3, indicating a fairly linear increase in \(y\) as \(x\) increases around \(x=2\).
Derivatives at a point are a cornerstone of calculus. They allow us to understand the behavior of functions in a most granular and precise manner, providing essential insights into the dynamics of various real-world phenomena.
Other exercises in this chapter
Problem 3
Find \(f^{\prime}(x)\). $$f(x)=-4 x^{2} \cos x$$
View solution Problem 3
Compute the derivative of the given function \(f(x)\) by (a) multiplying and then differentiating and (b) using the product rule. Verify that (a) and (b) yield
View solution Problem 4
Let \(f(x)=5 \sqrt{x}\) and \(g(x)=4+\cos x\) (a) Find \((f \circ g)(x)\) and \((f \circ g)^{\prime}(x)\) (b) Find \((g \circ f)(x)\) and \((g \circ f)^{\prime}
View solution Problem 4
Find \(d y / d x\) $$y=\frac{1}{2}\left(x^{4}+7\right)$$
View solution