A Curve Is Given By The Equation: $y = 5x^3 - 7x^2 + 3x + 2$. Find The Gradient Of The Curve At $x = 1$.

Introduction

In calculus, the gradient of a curve is a measure of how steep the curve is at a given point. It is an essential concept in mathematics, with applications in physics, engineering, and economics. In this article, we will explore how to find the gradient of a curve given by the equation $y = 5x^3 - 7x^2 + 3x + 2$ at $x = 1$.

What is the Gradient of a Curve?

The gradient of a curve is a measure of how steep the curve is at a given point. It is defined as the derivative of the curve with respect to the variable $x$. In other words, it is the rate of change of the curve with respect to $x$. The gradient is denoted by the symbol $\frac{dy}{dx}$.

Finding the Derivative of the Curve

To find the gradient of the curve, we need to find the derivative of the curve with respect to $x$. The derivative of a curve is found using the power rule of differentiation, which states that if $y = x^n$, then $\frac{dy}{dx} = nx^{n-1}$.

Using this rule, we can find the derivative of the curve $y = 5x^3 - 7x^2 + 3x + 2$.

Step 1: Differentiate the Term $5x^3$

Using the power rule of differentiation, we can differentiate the term $5x^3$ as follows:

$$\frac{d}{dx}(5x^3) = 5 \cdot 3x^{3-1} = 15x^2$$

Step 2: Differentiate the Term $-7x^2$

Using the power rule of differentiation, we can differentiate the term $-7x^2$ as follows:

$$\frac{d}{dx}(-7x^2) = -7 \cdot 2x^{2-1} = -14x$$

Step 3: Differentiate the Term $3x$

Using the power rule of differentiation, we can differentiate the term $3x$ as follows:

$$\frac{d}{dx}(3x) = 3 \cdot 1x^{1-1} = 3$$

Step 4: Differentiate the Term $2$

The derivative of a constant is zero, so we can differentiate the term $2$ as follows:

$$\frac{d}{dx}(2) = 0$$

Step 5: Combine the Derivatives

Now that we have found the derivatives of each term, we can combine them to find the derivative of the curve.

$$\frac{dy}{dx} = 15x^2 - 14x + 3$$

Finding the Gradient at $x = 1$

Now that we have found the derivative of the curve, we can find the gradient at $x = 1$ by substituting $x = 1$ into the derivative.

$$\frac{dy}{dx} = 15(1)^2 - 14(1) + 3 = 15 - 14 + 3 = 4$$

Therefore, the gradient of the curve at $x = 1$ is $4$.

Conclusion

In this article, we have explored how to find the gradient of a curve given by the equation $y = 5x^3 - 7x^2 + 3x + 2$ at $x = 1$. We have used the power rule of differentiation to find the derivative of the curve, and then substituted $x = 1$ into the derivative to find the gradient. The gradient of the curve at $x = 1$ is $4$.

Further Reading

If you are interested in learning more about calculus and the gradient of a curve, I recommend checking out the following resources:

• Calculus for Dummies
• The Gradient of a Curve
• Calculus: Early Transcendentals

Glossary

• Gradient: A measure of how steep a curve is at a given point.
• Derivative: The rate of change of a curve with respect to the variable $x$.
• Power Rule of Differentiation: A rule that states that if $y = x^n$, then $\frac{dy}{dx} = nx^{n-1}$.

References

• Calculus for Dummies
• The Gradient of a Curve
• Calculus: Early Transcendentals
  A Curve is Given by the Equation: $y = 5x^3 - 7x^2 + 3x + 2$. Find the Gradient of the Curve at $x = 1$. Q&A =====================================================================================================

Introduction

In our previous article, we explored how to find the gradient of a curve given by the equation $y = 5x^3 - 7x^2 + 3x + 2$ at $x = 1$. In this article, we will answer some frequently asked questions about the gradient of a curve and provide additional information to help you understand this concept better.

Q: What is the gradient of a curve?

A: The gradient of a curve is a measure of how steep the curve is at a given point. It is defined as the derivative of the curve with respect to the variable $x$. In other words, it is the rate of change of the curve with respect to $x$.

Q: How do I find the gradient of a curve?

A: To find the gradient of a curve, you need to find the derivative of the curve with respect to $x$. This can be done using the power rule of differentiation, which states that if $y = x^n$, then $\frac{dy}{dx} = nx^{n-1}$.

Q: What is the power rule of differentiation?

A: The power rule of differentiation is a rule that states that if $y = x^n$, then $\frac{dy}{dx} = nx^{n-1}$. This rule can be used to find the derivative of a curve with respect to $x$.

Q: How do I use the power rule of differentiation?

A: To use the power rule of differentiation, you need to identify the exponent of the variable $x$ in the equation of the curve. Then, you multiply the coefficient of the term by the exponent and subtract 1 from the exponent. This will give you the derivative of the curve with respect to $x$.

Q: What is the derivative of the curve $y = 5x^3 - 7x^2 + 3x + 2$?

A: To find the derivative of the curve $y = 5x^3 - 7x^2 + 3x + 2$, we can use the power rule of differentiation. The derivative of the curve is:

$$\frac{dy}{dx} = 15x^2 - 14x + 3$$

Q: How do I find the gradient of the curve at $x = 1$?

A: To find the gradient of the curve at $x = 1$, we can substitute $x = 1$ into the derivative of the curve. This will give us the gradient of the curve at $x = 1$.

Q: What is the gradient of the curve at $x = 1$?

A: The gradient of the curve at $x = 1$ is:

$$\frac{dy}{dx} = 15(1)^2 - 14(1) + 3 = 15 - 14 + 3 = 4$$

Q: What are some common applications of the gradient of a curve?

A: The gradient of a curve has many applications in physics, engineering, and economics. Some common applications include:

• Finding the maximum or minimum value of a function
• Determining the rate of change of a function
• Finding the equation of a tangent line to a curve
• Solving optimization problems

Conclusion

In this article, we have answered some frequently asked questions about the gradient of a curve and provided additional information to help you understand this concept better. We hope that this article has been helpful in clarifying any doubts you may have had about the gradient of a curve.

Further Reading

If you are interested in learning more about calculus and the gradient of a curve, we recommend checking out the following resources:

• Calculus for Dummies
• The Gradient of a Curve
• Calculus: Early Transcendentals

Glossary

• Gradient: A measure of how steep a curve is at a given point.
• Derivative: The rate of change of a curve with respect to the variable $x$.
• Power Rule of Differentiation: A rule that states that if $y = x^n$, then $\frac{dy}{dx} = nx^{n-1}$.

References

• Calculus for Dummies
• The Gradient of a Curve
• Calculus: Early Transcendentals