Suppose an object moves along the y axis so that its location is yequals= f (x) equals=xsquared2plus+x at time x (y is in meters, x is in seconds). Find (A) The average velocity (the average rate of change of y with respect to x) for x changing from 55 to 88. (B) The average velocity for x changing from 55 to 55plus+h. (C) The instantaneous velocity at xequals=55 second.

A) The average velocity is 144 m/s.

B) The average velocity is (111 + h) m/s.

C) The instantaneous velocity is 111 m/s.

Explanation:

The position of the object is given by the following function:

y = f (x) = x² + x

A) The average velocity can be calculated as follows:

av = (f (xf) - f (x0)) / (xf - x0)

Where:

av = average velocity

f (xf) = the value of the function at x = x-final (xf)

f (x0) = the value of the function at x = x-initial (x0)

xf = x-final, final value of "x"

x0 = x-initial, initial value of "x"

Then the averge rate of change from x0 = 55 to xf = 88 will be:

av = (f (88) - f (55)) / (88 - 55)

f (x) = x² + x

Evaluating the function in x = 88 and x = 55:

f (88) = 88² + 88 = 7832 m

f (55) = 55² + 55 = 3080 m

Then:

av = 7832 m - 3080 m / (88 s - 55 s) = 144 m/s

B) av = (f (55 + h) - f (55)) / (55 + h - 55)

Evaluating the function in x = 55 + h:

f (55+h) = (55+h) ² + (55+h)

f (55+h) = (55+h) · (55+h) + (55 + h)

f (55+h) = 55² + 55h + 55h + h² + 55 + h

Evaluating the function in x = 55

f (55) = 55² + 55

Then f (xf) - f (x0):

f (55+h) - f (55) = 55² + 55h + 55h + h² + 55 + h - 55² - 55

f (55+h) - f (55) = h (111 + h)

Then:

av = h (111 + h) / h = (111 + h) m/s (if h ≠ 0)

C) The instantaneous velocity is obtained by derivating the function and evaluating the derivative at x = 55.

Then:

f' (x) = 2x + 1

f' (55) = 2 · 55 + 1 = 111 m/s