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

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