Find an equation of the line tangent to the given curve at x =a. Use a graphing utility to graph the curve and the tangent line on the same set of axes. y =9+ 2x + 4x e X; a= 0
Arc Length
Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length.
Line Integral
A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.
Triple Integral
Examples:
![**Finding the Tangent Line Equation**
To find the equation of the line tangent to the given curve at \( x = a \), follow these steps. You will utilize a graphing utility to visualize the curve and the tangent line on the same set of axes.
**Given Equation:**
\[ y = 9 + 2x + 4x \cdot e^x \]
**Point of Tangency:**
\[ a = 0 \]
**Instructions:**
1. **Calculate the Derivative:**
- Determine the derivative of the function \( y = 9 + 2x + 4x \cdot e^x \).
- Use the product rule for the \( 4x \cdot e^x \) term.
2. **Evaluate the Derivative at \( x = 0 \):**
- This will give the slope of the tangent line at the point \( x = 0 \).
3. **Find the Tangent Line Equation:**
- Utilize the point-slope form of the line equation, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope found in the previous step, and \((x_1, y_1)\) is the point of tangency.
4. **Graphing:**
- Use a graphing utility to plot both the original curve and the tangent line.
- Ensure both plots are on the same axes for a clear comparison.
The goal is to visually see where the tangent line touches the curve and how it proceeds with the same slope at that point of contact.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2dedb04f-ec8e-4d86-87d9-6910be748f13%2F2a896209-ba3b-4759-97be-504a1d1f7017%2Fbbsnwn_processed.png&w=3840&q=75)
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