Let y = 10x^3+2x^2+5 a. find the average of change at which y changes with x over the interval [2,5]? b. find the instantaneous rate of change y with respect to x at the point x = -1?

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Answer 1 · 2018-03-17T14:28:35

See below.

The average rate of change for a function #f(x)# over an interval #[a,b]# is given by:

#(f(b)-f(a))/(b-a)#

We have:

#f(x)=10x^3+2x^2+5#

#a=2 , b=5#

#((10(5)^3+2(5)^2+5)-(10(2)^3+2(2)^2+5))/(5-2)#

#((1305)-(93))/(3)=color(blue)(404)#

The instantaneous rate of change of #y# in respect of #x# at a point #x=a# is:

#f'(a)#

This means we first need to find the derivative of #f(x)#

Using the power rule:

#dy/dx(ax^n)=anx^(n-1)#

This is distributive over the sum:

#dy/dx(ax^2+bx+c)=dy/dx(ax^2)+dy/dx(bx)+dy/dx(c)#

#dy/dx(10x^3+2x^2+5)=30x^2+4x#

#dy/dx=30x^2+4x#

Plugging in #\ \ \ \x=-1#

#30(-1)^2+4(-1)=color(blue)(26)#