given function f(X)=X^3+3x
findthe rate of change between the two stated values for x:1 to 2
find the equation of a secant line (1,f(1)and(2,f(2))
f(1) = 4
f(2) = 14
rate of change = slope = 10
line containing (1,4) with slope 10 is
(y-4) = 10(x-1)
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To find the rate of change between two values of x, we need to calculate the difference in the function values and divide it by the difference in the x-values.
1. Find f(1):
Substitute x=1 into the function f(x):
f(1) = (1)^3 + 3(1)
= 1 + 3
= 4
2. Find f(2):
Substitute x=2 into the function f(x):
f(2) = (2)^3 + 3(2)
= 8 + 6
= 14
3. Calculate the rate of change:
Rate of change = (f(2) - f(1)) / (2 - 1)
= (14 - 4) / 1
= 10
Therefore, the rate of change between x=1 and x=2 is 10.
To find the equation of a secant line, which passes through two points on the graph of a function, we will use the point-slope form of a linear equation.
Given the points (1, f(1)) = (1, 4) and (2, f(2)) = (2, 14), we can find the slope of the secant line:
Slope = (f(2) - f(1)) / (2 - 1)
= (14 - 4) / (2 - 1)
= 10
Using the point-slope form, we can now find the equation of the secant line:
y - y₁ = m(x - x₁)
where (x₁, y₁) is one of the given points, and m is the slope.
Using the point (1, 4):
y - 4 = 10(x - 1)
Simplifying the equation:
y = 10x - 10 + 4
y = 10x - 6
Therefore, the equation of the secant line passing through the points (1, 4) and (2, 14) is y = 10x - 6.