given f(x)=x^3+5x find the rate of change for the interval [0.5,0.51]
find
f(.51) and f(.5)
evaluate: ( f(.51 - f(.5) )/(.51 - .5)
To find the rate of change for the interval [0.5, 0.51] for the function f(x) = x^3 + 5x, we need to calculate the average rate of change.
The average rate of change is given by the difference in function values divided by the difference in x-values.
First, let's evaluate the function at the lower and upper limits of the interval:
For x = 0.5:
f(0.5) = (0.5)^3 + 5(0.5) = 0.125 + 2.5 = 2.625
For x = 0.51:
f(0.51) = (0.51)^3 + 5(0.51) = 0.132651 + 2.55 = 2.682651
Now, we can calculate the rate of change:
Rate of change = (f(0.51) - f(0.5)) / (0.51 - 0.5)
= (2.682651 - 2.625) / (0.51 - 0.5)
= 0.057651 / 0.01
= 5.7651
Therefore, the rate of change for the interval [0.5, 0.51] for the function f(x) = x^3 + 5x is 5.7651.
To find the rate of change for the interval [0.5, 0.51] for the given function f(x) = x^3 + 5x, we need to calculate the average rate of change.
The average rate of change of a function over an interval is determined by taking the difference between the function values at the endpoints of the interval and dividing it by the difference between the input values of those endpoints.
Let's calculate the average rate of change step by step:
Step 1: Calculate f(0.5) and f(0.51)
Plug in x = 0.5 into the function f(x) = x^3 + 5x:
f(0.5) = (0.5)^3 + 5(0.5)
Simplify:
f(0.5) = 0.125 + 2.5
f(0.5) = 2.625
Now plug in x = 0.51 into the function f(x) = x^3 + 5x:
f(0.51) = (0.51)^3 + 5(0.51)
Simplify:
f(0.51) = 0.132651 + 2.55
f(0.51) = 2.682651
Step 2: Calculate the difference in function values: f(0.51) - f(0.5)
f(0.51) - f(0.5) = 2.682651 - 2.625
f(0.51) - f(0.5) = 0.057651
Step 3: Calculate the difference in input values: 0.51 - 0.5
0.51 - 0.5 = 0.01
Step 4: Calculate the average rate of change:
Average rate of change = (f(0.51) - f(0.5)) / (0.51 - 0.5)
Plug in the values we calculated:
Average rate of change = 0.057651 / 0.01
Average rate of change = 5.7651
Therefore, the rate of change for the interval [0.5, 0.51] for the function f(x) = x^3 + 5x is 5.7651.