f(x)=4x+3 g(x) = 1/2x^2+2
We have to find the rate of change of each function from x=0 to x=1. Before starting the computations, let's remember the expression that gives us the rate of change of a function F(x) between x=a and x=b.
Rate of Change &= Change iny/Change in x
⇕
Rate of Change&= F(b)-F(a)/b-a
Using this formula, we can find the rate of change of each of the given functions.
Rate of Change = F(b)-F(a)/b-a
f(1)- f(0)/1-0
g(1)- g(0)/1-0
4(1)+3-(4(0)+3)/1-0
12(1)^2+2-( 12(0)^2+2)/1-0
4
0.5
From the table above, we conclude that the function f(x) has a greater rate of change from x=0 to x=1. We can also check it graphically.
Looking at the graph, we can say that both functions have the same change in the x-coordinates, but the change in the y-coordinates is greater for the function f(x). Therefore, f(x) has a greater rate of change.
b First, let's remember the formula to calculate the rate of change of a function F(x) from x=a to x=b.
Rate of Change &= Change iny/Change in x
⇕
Rate of Change&= F(b)-F(a)/b-a
As in Part A, let's find the rate of change of each of the given functions from x=2 and x=3.
Rate of Change = F(b)-F(a)/b-a
f(3)- f(2)/3-2
g(3)- g(2)/3-2
4(3)+3-(4(2)+3)/3-2
12(3)^2+2-( 12(2)^2+2)/3-2
4
2.5
From the table above, we conclude once more that the function f(x) has a greater rate of change from x=2 to x=3. We can also check it graphically.
Looking at the graph, we can say that both functions have the same change in x-coordinates, but the change in y-coordinates is greater for the function f(x). Therefore, f(x) has a greater rate of change.
c From the two previous results, and since the graph of y= f(x) is a line, we can affirm that the rate of change of f(x) is always constant and equal to 4. However, the rate of change of g(x) is not constant, and it increases as x increases.
Rate of Change = g(b)- g(a)/b-a, a< bWe are interested in finding, if any, the values of a and b for which the expression above is greater than 4.
