Consider The Function F ( X ) = X 2 + X F(x)=x^2+x F ( X ) = X 2 + X And The Point ( 1 , 2 (1,2 ( 1 , 2 ].(a) Find The Slope Of The Tangent To The Graph Of F ( X F(x F ( X ] At Any Point. □ \square □ (b) Find The Slope Of The Tangent At The Given Point. □ \square □ (c) Write

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In this article, we will delve into the world of calculus and explore the concept of finding the slope of the tangent to a graph. We will consider the function f(x)=x2+xf(x) = x^2 + x and the point (1,2)(1, 2), and use this information to find the slope of the tangent at any point and at the given point.

What is the Slope of the Tangent?

The slope of the tangent to a graph at a given point is a measure of how steep the graph is at that point. It is calculated by finding the derivative of the function, which represents the rate of change of the function with respect to the variable.

Finding the Derivative of the Function

To find the derivative of the function f(x)=x2+xf(x) = x^2 + x, we will use the power rule of differentiation. The power rule states that if f(x)=xnf(x) = x^n, then f′(x)=nxn−1f'(x) = nx^{n-1}.

Using this rule, we can find the derivative of the function f(x)=x2+xf(x) = x^2 + x as follows:

f′(x)=ddx(x2+x)f'(x) = \frac{d}{dx} (x^2 + x) f′(x)=ddx(x2)+ddx(x)f'(x) = \frac{d}{dx} (x^2) + \frac{d}{dx} (x) f′(x)=2x+1f'(x) = 2x + 1

Therefore, the derivative of the function f(x)=x2+xf(x) = x^2 + x is f′(x)=2x+1f'(x) = 2x + 1.

Finding the Slope of the Tangent at Any Point

Now that we have found the derivative of the function, we can use it to find the slope of the tangent at any point. The slope of the tangent at any point is given by the value of the derivative at that point.

Let xx be any point on the graph of the function. Then, the slope of the tangent at that point is given by:

m=f′(x)=2x+1m = f'(x) = 2x + 1

Therefore, the slope of the tangent at any point is given by the equation m=2x+1m = 2x + 1.

Finding the Slope of the Tangent at the Given Point

Now, we will find the slope of the tangent at the given point (1,2)(1, 2). To do this, we will substitute the value of xx into the equation for the slope of the tangent.

Let x=1x = 1. Then, the slope of the tangent at the point (1,2)(1, 2) is given by:

m=f′(1)=2(1)+1m = f'(1) = 2(1) + 1 m=2+1m = 2 + 1 m=3m = 3

Therefore, the slope of the tangent at the point (1,2)(1, 2) is m=3m = 3.

Conclusion

In this article, we have explored the concept of finding the slope of the tangent to a graph. We have considered the function f(x)=x2+xf(x) = x^2 + x and the point (1,2)(1, 2), and used this information to find the slope of the tangent at any point and at the given point. We have shown that the slope of the tangent at any point is given by the equation m=2x+1m = 2x + 1, and that the slope of the tangent at the point (1,2)(1, 2) is m=3m = 3.

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Calculus, 2nd edition, James Stewart

Discussion

What do you think about the concept of finding the slope of the tangent to a graph? Do you have any questions or comments about this article? Please feel free to share your thoughts in the comments section below.

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In this article, we will answer some of the most frequently asked questions about finding the slope of the tangent to a graph. We will cover topics such as the definition of the slope of the tangent, how to find the derivative of a function, and how to use the derivative to find the slope of the tangent.

Q: What is the slope of the tangent?

A: The slope of the tangent to a graph at a given point is a measure of how steep the graph is at that point. It is calculated by finding the derivative of the function, which represents the rate of change of the function with respect to the variable.

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

A: To find the derivative of a function, you can use the power rule of differentiation. The power rule states that if f(x)=xnf(x) = x^n, then f′(x)=nxn−1f'(x) = nx^{n-1}. You can also use other rules of differentiation, such as the product rule and the quotient rule, to find the derivative of a function.

Q: What is the power rule of differentiation?

A: The power rule of differentiation states that if f(x)=xnf(x) = x^n, then f′(x)=nxn−1f'(x) = nx^{n-1}. This means that if you have a function of the form f(x)=xnf(x) = x^n, you can find its derivative by multiplying the function by nn and then reducing the exponent by 1.

Q: How do I use the derivative to find the slope of the tangent?

A: To use the derivative to find the slope of the tangent, you need to substitute the value of xx into the derivative. This will give you the slope of the tangent at that point.

Q: What is the difference between the slope of the tangent and the slope of the secant?

A: The slope of the tangent is a measure of how steep the graph is at a given point, while the slope of the secant is a measure of how steep the graph is between two points. The slope of the tangent is always equal to the slope of the secant at the point where the tangent touches the graph.

Q: Can I find the slope of the tangent at any point on the graph?

A: Yes, you can find the slope of the tangent at any point on the graph. To do this, you need to substitute the value of xx into the derivative. This will give you the slope of the tangent at that point.

Q: What is the significance of the slope of the tangent?

A: The slope of the tangent is an important concept in calculus, as it represents the rate of change of the function with respect to the variable. It is used to find the maximum and minimum values of a function, as well as the points of inflection.

Q: Can I use the slope of the tangent to find the maximum and minimum values of a function?

A: Yes, you can use the slope of the tangent to find the maximum and minimum values of a function. To do this, you need to find the critical points of the function, which are the points where the slope of the tangent is equal to zero.

Q: What is the relationship between the slope of the tangent and the second derivative?

A: The slope of the tangent is related to the second derivative of the function. The second derivative represents the rate of change of the first derivative, which is the slope of the tangent.

Conclusion

In this article, we have answered some of the most frequently asked questions about finding the slope of the tangent to a graph. We have covered topics such as the definition of the slope of the tangent, how to find the derivative of a function, and how to use the derivative to find the slope of the tangent. We hope that this article has been helpful in understanding the concept of the slope of the tangent.

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Calculus, 2nd edition, James Stewart

Discussion

Do you have any questions or comments about this article? Please feel free to share your thoughts in the comments section below.

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