The problem provided is as follows:Let $\vec{F}(x, y) = (x+1) \vec{i}$, and let $C$ be the portion of the parabola $y = x^2$ that starts at $(0, 0)$ and ends at $(2, 4)$.(a) Without doing any calculations, state whether $\int_{C} \vec{F} \cdot d\vec{r}$ is positive, negative, or zero. Clearly state your answer, and explain based on the diagram to the right how you know your answer is correct.**Explanation of the diagram:**The diagram shows the curve $C$, which is part of the parabola $y = x^2$, ranging from $(0, 0)$ to $(2, 4)$ on the xy-plane. The curve is plotted against a field of vectors (depicted as arrows) pointing upward. These vectors, $\vec{F}(x, y) = (x+1) \vec{i}$, are aligned along the positive x-direction and increase in magnitude as x increases.Since the vectors are always horizontal (along the x-axis) and the curve progresses from left to right while increasing in both x and y, the dot product $\vec{F} \cdot d\vec{r}$ will be positive. This is because the direction of $\vec{F}$ and the tangent vector along the curve are in the same general direction, resulting in a positive dot product.Therefore, $\int_{C} \vec{F} \cdot d\vec{r}$ is positive. The alignment of the vectors with the direction of movement along the curve confirms this conclusion visually.

Answered: Image /qna-images/answer/6dd79a70-e6fa-45f5-881a-2ef0af6f7e62.jpg

Let F(r, y) = (x + 1)7, and let C be the portion of the parabola y = x² that starts at (0,0) and ends at (2, 4). (a) Without doing any calculations, state whether fF•dř° is positive, neg- ative, or zero. Clearly state your answer, and explain based on the diagram to the right how you know your answer is correct.

Let F(r, y) = (x + 1)7, and let C be the portion of the parabola y = x² that starts at (0,0) and ends at (2, 4). (a) Without doing any calculations, state whether fF•dř° is positive, neg- ative, or zero. Clearly state your answer, and explain based on the diagram to the right how you know your answer is correct.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Find the standard equation of the parabola with x intercepts at x = 2 and x -3, and a y-intercept at…

A: Solution is given below

Q: The focus of the parabola defined by the equation x2 = 20 y is %3D (4, 5) (0, 0) O A. В. (0, 5)| (0,…

Q: 7. A parabola whose equation is y =r² - 2x + k has a turning point with coordinates (1,-5). Find the…

A: We have to find k

Q: Please don't provide handwritten solution...

A: Explanation is given below:Explanation:Given equation: 5y−x2=20 Rewrite it in the standard form of a…

Q: Find the equation for the parabola that has its focus at the (−7/4,−9) and has directrix at x=15/4

A: Here we have to find equation of parabola with given focus and directrix.

Q: You will transform the basic (parent) quadratic relation y =x² . Your transformation needs to…

Q: Find all points where the parabola y = 9 - x² 1 intersects the line y = zx+6. 2

Q: 3. Complete the table for each parabola Equation Y= -0.4(x-2)2 Y = 3(x + 5) 2 + 3 Direction of…

Q: If the vertex of the parabola f (x)= x² +bx+c is at (7, -4), find b and c. (Note: a = 1).

Q: Q2. Verify Stoke's theorem for the function F = z²(cosø +e, over the rectangular closed contour…

A: Given, F→=z2cosφ+12eρ→ As we know that, ∮F→·dl→=∫abF→·dl→+∫bcF→·dl→+∫cdF→·dl→+∫daF→·dl→ As, F→…

Q: I am on a planet whose acceleration due to gravity is g=6 ft/s2. I'm standing on a cliff edge with…

A: Consider the equation

Q: 3. Write the equation of the parabola that matches each description. a) The graph of y =x² is…

A: 3. According to questionGiven Curve :y = x2

Q: 5. Show that the sum of the intercepts of the tangent to the parabola (Art. 129), x+y=a¹, is equal…

A: We need to prove given theorem.

Q: For what values of a and b is the 2x + y =b line tangent tothe parabola y =ax2 when x =2?

A: Rewrite the equation of the tangent line as y=-2x+b and its slope is -2.

Question

The problem provided is as follows:

Let $\vec{F}(x, y) = (x+1) \vec{i}$, and let $C$ be the portion of the parabola $y = x^2$ that starts at $(0, 0)$ and ends at $(2, 4)$.

(a) Without doing any calculations, state whether $\int_{C} \vec{F} \cdot d\vec{r}$ is positive, negative, or zero. Clearly state your answer, and explain based on the diagram to the right how you know your answer is correct.

**Explanation of the diagram:**

The diagram shows the curve $C$, which is part of the parabola $y = x^2$, ranging from $(0, 0)$ to $(2, 4)$ on the xy-plane. The curve is plotted against a field of vectors (depicted as arrows) pointing upward. These vectors, $\vec{F}(x, y) = (x+1) \vec{i}$, are aligned along the positive x-direction and increase in magnitude as x increases.

Since the vectors are always horizontal (along the x-axis) and the curve progresses from left to right while increasing in both x and y, the dot product $\vec{F} \cdot d\vec{r}$ will be positive. This is because the direction of $\vec{F}$ and the tangent vector along the curve are in the same general direction, resulting in a positive dot product.

Therefore, $\int_{C} \vec{F} \cdot d\vec{r}$ is positive. The alignment of the vectors with the direction of movement along the curve confirms this conclusion visually.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Given a vector field

VIEW

Note:

VIEW

From the given graph, we can conclude:

VIEW

Step by step

Solved in 3 steps with 4 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Implicit Differentiation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Express your final answer in Mdx + Ndy = 0 1. Parabola whose vertex is at point P(13, 60).
Classify the following parabola as horizontal and vertical, indicate where they open.
The parabola y = x + 9 has two tangents which pass through the point (0, – 6). One is tangent to the to the parabola at (A, A² + 9) and the other at (- A, A? + 9) Find (the positive number) A.
Find the shortest distance of the point (10,4) to the parabola y=x2
Calculate y′′ for the parabola equation y2=4x.
Solve and show work for #9