I don't understand these types of problems-
Find the average rate of change of f between 1 and x
f(x)-f(1)/x-1
and X cannot be equal to 1
1) f(x)= the square root of x+3
2) f(x)=1/(x^2)
3) f(x)=x^3 + x
To find the average rate of change of a function between two points, you need to calculate the difference in function values divided by the difference in the corresponding input values. In this case, you're given the specific function forms and asked to find the average rate of change between 1 and x.
Let's go through each problem step by step:
1) f(x) = √(x + 3):
To find the average rate of change between 1 and x for this function, substitute the given function into the formula:
Average rate of change = (f(x) - f(1)) / (x - 1)
Replace f(x) with √(x + 3) and f(1) with √(1 + 3):
Average rate of change = (√(x + 3) - √(1 + 3)) / (x - 1)
Simplify the numerator:
Average rate of change = (√(x + 3) - 2) / (x - 1)
2) f(x) = 1 / (x^2):
Again, substitute the given function into the formula:
Average rate of change = (f(x) - f(1)) / (x - 1)
Replace f(x) with 1 / (x^2) and f(1) with 1 / (1^2):
Average rate of change = (1 / (x^2) - 1 / 1) / (x - 1)
Simplify the numerator:
Average rate of change = (1 / (x^2) - 1) / (x - 1)
3) f(x) = x^3 + x:
Once more, substitute the given function into the formula:
Average rate of change = (f(x) - f(1)) / (x - 1)
Replace f(x) with x^3 + x and f(1) with 1^3 + 1:
Average rate of change = ((x^3 + x) - (1^3 + 1)) / (x - 1)
Simplify the numerator:
Average rate of change = (x^3 + x - 2) / (x - 1)
Note that in all cases, x cannot be equal to 1, as specified in the problem statement.