Question

Suppose that an object moves along the y-axis so that its location is y=x2+3x at time x. (Here y is in meters and x is in seconds.)

a. Find the average velocity (the average rate of change of y with respect to x) for x changing from 3 to 7.
b. Find the average velocity for xx changing from 66 to 6+h6+h seconds.
c. Find the instantaneous velocity at x=6x=6 seconds (include units help (units)).

Answer 1

a) 13 m/s

b) (15 + h) m/s

c) 15 m/s

Explanation:

if the location is

y=x²+3*x

then the average velocity from 3 to 7 is

Δy/Δx=[y(7)-y(3)]/(7-3)=[7²+3*7- (3²+3*3)]/4= 13 m/s

then the average velocity from x=6 to to x=6+h

Δy/Δx=[y(6+h)-y(6)]/(6+h-6)=[(6+h)²+3*(6+h)- (6²+3*6)]/h= (2*6*h+3*h+h²)/h=2*6+3= (15 + h) m/s

the instantaneous velocity can be found taking the limit of Δy/Δx when h→0. Then

when h→0 , limit Δy/Δx= (15 + h) m/s = 15 m/s

then v= 15 m/s

also can be found taking the derivative of y in x=6

v=dy/dx=2*x+3

for x=6

v=dy/dx=2*6+3 = 12+3=15 m/s