The graphs of the function f and its derivative f′ are shown above for −1≤x≤4. (a) Find the average rate of change of f over the interval −1≤x≤4. For how many values of x in the interval −1≤x≤4 does the instantaneous rate of change of f equal the average rate of change of f over that interval?

(b) Write an equation for the line tangent to the graph of f at x=1.

(c) For each of limx→2f(x)−f(2)x−2 and limx→3f(x)−f(3)x−3, find the value or give a reason why it does not exist.

(d) Let g be the function defined by g(x)=exf(x). Find g′(0).

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is