Find An Equation Of The Line Tangent To The Graph Of F ( X ) = − 3 − 7 X 2 F(x) = -3 - 7x^2 F ( X ) = − 3 − 7 X 2 At The Point { (5, -178)$}$.

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Introduction

In calculus, finding the equation of a tangent line to a curve is a fundamental concept that has numerous applications in various fields, including physics, engineering, and economics. The tangent line to a curve at a given point is the line that just touches the curve at that point and has the same slope as the curve at that point. In this article, we will discuss how to find the equation of the tangent line to the graph of the function $f(x) = -3 - 7x^2$ at the point $(5, -178)$.

Understanding the Function

The given function is $f(x) = -3 - 7x^2$. This is a quadratic function, which means it has a parabolic shape. The graph of this function is a downward-facing parabola, since the coefficient of the $x^2$ term is negative. The vertex of the parabola is at the point $(0, -3)$, which is the minimum point of the function.

Finding the Derivative

To find the equation of the tangent line, we need to find the derivative of the function. The derivative of a function represents the rate of change of the function with respect to the variable. In this case, we need to find the derivative of $f(x) = -3 - 7x^2$.

Using the power rule of differentiation, which states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$, we can find the derivative of $f(x) = -3 - 7x^2$.

$$f'(x) = -7(2x) = -14x$$

Finding the Slope of the Tangent Line

The slope of the tangent line to the graph of $f(x)$ at the point $(a, f(a))$ is given by the derivative of the function evaluated at $x = a$. In this case, we need to find the slope of the tangent line at the point $(5, -178)$.

Substituting $x = 5$ into the derivative, we get:

$$f'(5) = -14(5) = -70$$

Finding the Equation of the Tangent Line

Now that we have the slope of the tangent line, we can use the point-slope form of a linear equation to find the equation of the tangent line. The point-slope form of a linear equation 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.

In this case, we have the point $(5, -178)$ and the slope $m = -70$. Substituting these values into the point-slope form, we get:

$$y - (-178) = -70(x - 5)$$

Simplifying this equation, we get:

$$y + 178 = -70x + 350$$

Subtracting 178 from both sides, we get:

$$y = -70x + 172$$

Conclusion

In this article, we discussed how to find the equation of the tangent line to the graph of the function $f(x) = -3 - 7x^2$ at the point $(5, -178)$. We found the derivative of the function, which represents the rate of change of the function with respect to the variable. We then used the point-slope form of a linear equation to find the equation of the tangent line. The final equation of the tangent line is $y = -70x + 172$.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Find the derivative of the function $f(x) = -3 - 7x^2$.
2. Evaluate the derivative at $x = 5$ to find the slope of the tangent line.
3. Use the point-slope form of a linear equation to find the equation of the tangent line.
4. Simplify the equation to get the final answer.

Common Mistakes

Here are some common mistakes to avoid when solving this problem:

• Not finding the derivative of the function.
• Not evaluating the derivative at the correct point.
• Not using the point-slope form of a linear equation to find the equation of the tangent line.
• Not simplifying the equation to get the final answer.

Real-World Applications

The concept of finding the equation of a tangent line has numerous real-world applications, including:

• Physics: The tangent line to a curve can represent the velocity of an object at a given point in time.
• Engineering: The tangent line to a curve can represent the slope of a road or a building.
• Economics: The tangent line to a curve can represent the rate of change of a quantity with respect to a variable.

Final Answer

The final answer is $y = -70x + 172$.

Introduction

In our previous article, we discussed how to find the equation of the tangent line to the graph of the function $f(x) = -3 - 7x^2$ at the point $(5, -178)$. In this article, we will answer some common questions related to finding the equation of the tangent line to a curve.

Q: What is the tangent line to a curve?

A: The tangent line to a curve at a given point is the line that just touches the curve at that point and has the same slope as the curve at that point.

Q: How do I find the equation of the tangent line to a curve?

A: To find the equation of the tangent line to a curve, you need to find the derivative of the function, evaluate the derivative at the point of tangency, and use the point-slope form of a linear equation to find the equation of the tangent line.

Q: What is the point-slope form of a linear equation?

A: The point-slope form of a linear equation 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.

Q: How do I find the derivative of a function?

A: To find the derivative of a function, you need to use the power rule of differentiation, which states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

Q: What is the significance of the derivative in finding the equation of the tangent line?

A: The derivative represents the rate of change of the function with respect to the variable. It is used to find the slope of the tangent line to the curve at a given point.

Q: Can I use the equation of the tangent line to find the slope of the curve at a given point?

A: Yes, you can use the equation of the tangent line to find the slope of the curve at a given point. The slope of the tangent line is equal to the derivative of the function evaluated at the point of tangency.

Q: What are some common mistakes to avoid when finding the equation of the tangent line?

A: Some common mistakes to avoid when finding the equation of the tangent line include:

• Not finding the derivative of the function.
• Not evaluating the derivative at the correct point.
• Not using the point-slope form of a linear equation to find the equation of the tangent line.
• Not simplifying the equation to get the final answer.

Q: What are some real-world applications of finding the equation of the tangent line?

A: Some real-world applications of finding the equation of the tangent line include:

• Physics: The tangent line to a curve can represent the velocity of an object at a given point in time.
• Engineering: The tangent line to a curve can represent the slope of a road or a building.
• Economics: The tangent line to a curve can represent the rate of change of a quantity with respect to a variable.

Q: Can I use the equation of the tangent line to find the equation of the curve?

A: No, you cannot use the equation of the tangent line to find the equation of the curve. The equation of the tangent line is a linear equation that represents the slope of the curve at a given point, while the equation of the curve is a nonlinear equation that represents the relationship between the variables.

Q: What is the final answer to the problem of finding the equation of the tangent line to the graph of $f(x) = -3 - 7x^2$ at the point $(5, -178)$?

A: The final answer is $y = -70x + 172$.

Conclusion

In this article, we answered some common questions related to finding the equation of the tangent line to a curve. We discussed the significance of the derivative in finding the equation of the tangent line, common mistakes to avoid, and real-world applications of finding the equation of the tangent line. We also provided a step-by-step solution to the problem of finding the equation of the tangent line to the graph of $f(x) = -3 - 7x^2$ at the point $(5, -178)$.