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Chapter 3: Problem 1

If the line \(y=7 x-4\) is tangent to \(f(x)\) at \(x=2,\) find \(f(2)\) and \(f^{\prime}(2)\).

Short Answer

Expert verified
The values of \(f(2)\) and \(f'(2)\) are 10 and 7 respectively.

Step by step solution

01

Find the derivative at the tangent point

Since \(f'(x)\) at \(x=2\) would be the same as the slope of the line tangent to \(f(x)\) at that point, and the slope of the line \(y=7x-4\) is 7, it follows that \(f'(2)=7\).
02

Find the value of the function at the tangent point

Since the line \(y=7x-4\) is tangent to \(f(x)\) at \(x=2\), we can put \(x=2\) into \(y=7x-4\) to get the y-coordinate of the point of tangency, which is \(f(2)\). Therefore, \(f(2) = 7*2 - 4 = 10\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Finding Derivative
The process of finding the derivative of a function at a specific point helps us identify the slope of the tangent line to the curve at that point. In calculus, the derivative of a function, denoted as \(f'(x)\), provides the rate at which the function is changing. When the exercise mentions that the line \(y=7x-4\) is tangent to \(f(x)\) at \(x=2\), this implies that the derivative at this point is equal to the slope of the tangent line.
  • The slope of a line is the coefficient of \(x\) in its equation. Here, it is 7.
  • Therefore, the derivative of \(f(x)\) at \(x=2\) is also 7, which means \(f'(2)=7\).
This is why the first step in the solution involves understanding that the derivative at the tangent point is equivalent to the slope of the tangent line.
Point of Tangency
The point of tangency is where a line just barely touches a curve, indicating that they have the same slope at that exact point. In this exercise, we know the line is tangent to the function at \(x=2\). Finding the point of tangency helps us understand both the location on the \(x\)-axis and the corresponding value on the \(y\)-axis.
  • At \(x=2\), the curve \(f(x)\) is touched by the line \(y=7x-4\).
  • This point on the curve is special because it's where the change in the function exactly matches the slope of the line.
It's essential because it not only tells us the derivative but also hints at the specific value that \(f(x)\) takes at this point, known as \(f(2)\).
Value of Function at Point
To find the value of the function at a particular point, simply substitute the given \(x\) value into the equation of the tangent line. This calculation yields the \(y\)-coordinate of the point of tangency.
  • In the problem, substitute \(x=2\) into \(y=7x-4\).
  • You get \(y=7(2)-4\), which results in \(y=10\).
This means \(f(2)=10\), confirming the function value where the tangent line meets the curve. This step ensures that you've correctly identified both the slope (or derivative) and the exact location of tangency on the actual function \(f(x)\).

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