The parent function, f(x) = 5^x, has been vertically compressed by a factor of one-half, shifted to the left three units and dow
n two unlts.
Choose the correct function to represent the transformation.
Choose the correct function to represent the transformation.
1 answer:
Answer: Choice C
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Explanation:
The 1/2 out front handles the vertical compression by 1/2. For example, if two points are vertically spaced by 10 units, then their new vertical distance is now (1/2)*10 = 5 units.
The x+3 in the exponent means "shift 3 units to the left". Effectively what's going on is that the old input x is now x+3, ie 3 units larger than before. This shifts the xy axis itself 3 units to the right. If we held the curve fixed in place while the xy axis moved like this, then it gives the illusion the curve is moving 3 units to the left.
Then finally the -2 at the very end shifts the curve down 2 units. This is because whatever the y coordinate is, we subtract 2 from it to do this vertical shifting. For example, the point (0,62.5) shifts 2 units down to (0,60.5)
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Check the picture below.
so the picture has a rectangle that is 8 units high and 12 units wide, and it has a couple of "empty" trapezoids, with a height of 5 and "bases" of 9 and 3.
now, if we just take the whole area of the rectangle and then subtract the area of those two trapezoids, what's leftover is the blue area.
Solution :
Along the edge
The parametric equation for is given :
Along edge
The curve here is a quarter circle with the radius 9. Therefore, the parametric equation with the domain is then given by :
Along edge
The parametric equation for is :
Now,
x = 9t, ⇒ dx = 9 dt
y = 0, ⇒ dy = 0
And
Then :
= 0
And
x = 0, ⇒ dx = 0
y = 9 t, ⇒ dy = 9 dt
Therefore,
= 0 + 0 + 0
Applying the Green's theorem
Here,
Therefore,
The vector field F is = is conservative.
