Question

Finding \(f\) from \(f^{\prime}\) Sketch the graph of \(f^{\prime}(x)=2 .\) Then sketch three possible graphs of \(f\)

Short answer

Question: Based on the given derivative \(f'(x) = 2\), what can be concluded about the original function \(f(x)\) and provide three possible graphs for the function. Answer: The original function \(f(x)\) is a linear function with a constant slope of 2. Three possible graphs for the function are: 1. \(f(x) = 2x\) 2. \(f(x) = 2x + 1\) 3. \(f(x) = 2x - 1\)

Step-by-step solution

Plot \(f'(x)\)

Given \(f'(x) = 2\), it's clear that the derivative is a constant function. A constant function has a horizontal line as its graph. So, plot a horizontal line on the coordinate plane with the equation \(y = 2\).

Analyze \(f'(x)\)

Since the derivative, \(f'(x)\), is always positive (equal to 2), it means that the original function \(f(x)\) is always increasing. Additionally, the derivative is constant, which means that \(f(x)\) has a constant rate of increase (i.e., its slope is constant). This information tells us that \(f(x)\) must be a linear function.

Sketch possible graphs of \(f(x)\)

We know that \(f(x)\) is a linear function with a constant slope of 2, which means that its general equation is \(f(x) = 2x + C\), where \(C\) is an arbitrary constant that represents the intercept on the y-axis. Varying the value of C will give us different possible graphs of \(f(x)\). Let's sketch three such graphs: 1. Graph of \(f(x) = 2x\): Set \(C = 0\). Plot the line with equation \(y = 2x\). It will pass through the origin \((0,0)\) and have a slope of 2. 2. Graph of \(f(x) = 2x + 1\): Set \(C = 1\). Plot the line with equation \(y = 2x + 1\). It will pass through the point \((0,1)\) and have a slope of 2. 3. Graph of \(f(x) = 2x - 1\): Set \(C = -1\). Plot the line with equation \(y = 2x - 1\). It will pass through the point \((0,-1)\) and have a slope of 2. So, by looking at the graph of \(f'(x)\) and understanding the relationship between a function and its derivative, we have been able to sketch three different possible graphs of the original function \(f(x)\).