The Tangent Line To \[$ Y = F(x) \$\] At \[$ (-1, 3) \$\] Passes Through The Point \[$ (3, 6) \$\]. Compute The Following:a.) \[$ F(-1) = \$\] \[$\square\$\]b.) \[$ F^{\prime}(-1) = \$\]
Introduction
In calculus, the tangent line to a function at a given point is a fundamental concept that helps us understand the behavior of the function at that point. Given a function { y = f(x) $}$ and a point { (-1, 3) $}$, we are asked to find the tangent line to the function at this point, which passes through the point { (3, 6) $}$. To do this, we need to compute the value of the function at the point { (-1, 3) $}$ and the derivative of the function at this point.
The Equation of the Tangent Line
The equation of the tangent line to a function { y = f(x) $}$ at a point { (x_0, f(x_0)) $}$ is given by:
{ y - f(x_0) = f^{\prime}(x_0)(x - x_0) $}$
where { f^{\prime}(x_0) $}$ is the derivative of the function at the point { x_0 $}$.
The Given Information
We are given that the tangent line to the function { y = f(x) $}$ at the point { (-1, 3) $}$ passes through the point { (3, 6) $}$. This means that the equation of the tangent line at the point { (-1, 3) $}$ is:
{ y - 3 = f^{\prime}(-1)(x + 1) $}$
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 of the line.
The Slope of the Tangent Line
The slope of the tangent line to the function at the point { (-1, 3) $}$ is given by:
{ f^{\prime}(-1) $}$
The Equation of the Tangent Line at the Point { (3, 6) $}$
Since the tangent line passes through the point { (3, 6) $}$, we can substitute these values into the equation of the tangent line:
{ 6 - 3 = f^{\prime}(-1)(3 + 1) $}$
Simplifying the equation, we get:
{ 3 = f^{\prime}(-1)(4) $}$
Solving for { f^{\prime}(-1) $}$
Dividing both sides of the equation by 4, we get:
{ f^{\prime}(-1) = \frac{3}{4} $}$
The Value of the Function at the Point { (-1, 3) $}$
To find the value of the function at the point { (-1, 3) $}$, we can substitute { x = -1 $}$ into the equation of the tangent line:
{ 3 = f^{\prime}(-1)(-1 + 1) $}$
Simplifying the equation, we get:
{ 3 = f^{\prime}(-1)(0) $}$
This means that the value of the function at the point { (-1, 3) $}$ is:
{ f(-1) = 3 $}$
Conclusion
In this article, we have computed the value of the function at the point { (-1, 3) $}$ and the derivative of the function at this point. We have also found the equation of the tangent line to the function at the point { (-1, 3) $}$ and the slope of the tangent line. The value of the function at the point { (-1, 3) $}$ is { f(-1) = 3 $}$ and the derivative of the function at this point is { f^{\prime}(-1) = \frac{3}{4} $}$.
References
- [1] Calculus, 3rd edition, Michael Spivak
- [2] Calculus, 2nd edition, James Stewart
- [3] Calculus, 1st edition, Michael Spivak
Discussion
The tangent line to a function at a given point is a fundamental concept in calculus that helps us understand the behavior of the function at that point. In this article, we have computed the value of the function at the point { (-1, 3) $}$ and the derivative of the function at this point. We have also found the equation of the tangent line to the function at the point { (-1, 3) $}$ and the slope of the tangent line. The value of the function at the point { (-1, 3) $}$ is { f(-1) = 3 $}$ and the derivative of the function at this point is { f^{\prime}(-1) = \frac{3}{4} $}$.
Related Topics
- The derivative of a function
- The equation of a tangent line
- The slope of a tangent line
- The point-slope form of a line
Tags
- Calculus
- Derivative
- Tangent line
- Slope
- Point-slope form
Introduction
In our previous article, we discussed the tangent line to a function at a given point and computed the value of the function at the point { (-1, 3) $}$ and the derivative of the function at this point. In this article, we will answer some frequently asked questions related to the tangent line to a function at a given point.
Q1: What is the tangent line to a function at a given point?
A1: The tangent line to a function { y = f(x) $}$ at a point { (x_0, f(x_0)) $}$ is a line that passes through the point { (x_0, f(x_0)) $}$ and has the same slope as the function at that point.
Q2: How do I find the equation of the tangent line to a function at a given point?
A2: To find the equation of the tangent line to a function { y = f(x) $}$ at a point { (x_0, f(x_0)) $}$, you need to find the derivative of the function at that point and use the point-slope form of a line.
Q3: What is the slope of the tangent line to a function at a given point?
A3: The slope of the tangent line to a function { y = f(x) $}$ at a point { (x_0, f(x_0)) $}$ is given by the derivative of the function at that point, { f^{\prime}(x_0) $}$.
Q4: How do I find the value of the function at a given point?
A4: To find the value of the function { y = f(x) $}$ at a point { (x_0, f(x_0)) $}$, you can substitute { x = x_0 $}$ into the equation of the function.
Q5: What is the point-slope form of a line?
A5: 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 of the line.
Q6: How do I find the equation of the tangent line to a function at a given point using the point-slope form?
A6: To find the equation of the tangent line to a function { y = f(x) $}$ at a point { (x_0, f(x_0)) $}$ using the point-slope form, you need to find the derivative of the function at that point and substitute the values into the point-slope form.
Q7: What is the relationship between the tangent line and the function?
A7: The tangent line to a function at a given point is a line that passes through the point and has the same slope as the function at that point. This means that the tangent line is a good approximation of the function at that point.
Q8: How do I use the tangent line to a function at a given point to make predictions about the function?
A8: You can use the tangent line to a function at a given point to make predictions about the function by using the equation of the tangent line to estimate the value of the function at nearby points.
Q9: What are some common applications of the tangent line to a function at a given point?
A9: The tangent line to a function at a given point has many applications in physics, engineering, and economics, such as modeling the behavior of a function, predicting the value of a function at nearby points, and optimizing functions.
Q10: How do I find the equation of the tangent line to a function at a given point using calculus?
A10: To find the equation of the tangent line to a function { y = f(x) $}$ at a point { (x_0, f(x_0)) $}$ using calculus, you need to find the derivative of the function at that point and use the point-slope form of a line.
Conclusion
In this article, we have answered some frequently asked questions related to the tangent line to a function at a given point. We have discussed the definition of the tangent line, how to find the equation of the tangent line, and how to use the tangent line to make predictions about the function. We have also discussed some common applications of the tangent line to a function at a given point.
References
- [1] Calculus, 3rd edition, Michael Spivak
- [2] Calculus, 2nd edition, James Stewart
- [3] Calculus, 1st edition, Michael Spivak
Discussion
The tangent line to a function at a given point is a fundamental concept in calculus that helps us understand the behavior of the function at that point. In this article, we have discussed the definition of the tangent line, how to find the equation of the tangent line, and how to use the tangent line to make predictions about the function. We have also discussed some common applications of the tangent line to a function at a given point.
Related Topics
- The derivative of a function
- The equation of a tangent line
- The slope of a tangent line
- The point-slope form of a line
Tags
- Calculus
- Derivative
- Tangent line
- Slope
- Point-slope form