B and C are correct, A and D are incorrect
Negative 3x over the power by four
We want to find 
such that 
. This means
Integrating both sides of the latter equation with respect to 
tells us
and differentiating with respect to 
gives
Integrating both sides with respect to gives
Then
and differentiating both sides with respect to 
gives
So the scalar potential function is
By the fundamental theorem of calculus, the work done by 
along any path depends only on the endpoints of that path. In particular, the work done over the line segment (call it 
) in part (a) is
and does the same amount of work over both of the other paths.
In part (b), I don't know what is meant by "df/dt for F"...
In part (c), you're asked to find the work over the 2 parts (call them 
and 
) of the given path. Using the fundamental theorem makes this trivial:
Answer:
Below.
Step-by-step explanation:
I'll do number 20, 21, 22 for you.
20. It is ±1, ±2, ±4, ±8.
21. Reflected over the x-axis and down 4.
22. Common ratio = 25/5 = 5.
23. Imaginary roots occur in pairs so there is also a root -5i.
The answer is c.
19. Option (a).
This is a translation of 4 to the left ( due to the (x + 4).
