Given the function f(x) = x^3+ 3x
Find the rate of change between the two stated values for x: 1 to 2
Find the equation of a secant line containing the given points: (1, f(1)) and (2, f(2))
Find f(1) and f(2)
Find the slope f(1) - f(2) / 1 - 2
Write your equation
Rate of change = slope
To find the rate of change between two values of x, you calculate the difference in the function values divided by the difference in the x-values.
1. Calculate the function values:
For x = 1:
f(1) = (1)^3 + 3(1) = 1 + 3 = 4
For x = 2:
f(2) = (2)^3 + 3(2) = 8 + 6 = 14
2. Calculate the difference in function values:
Difference in function values = f(2) - f(1) = 14 - 4 = 10
3. Calculate the difference in x-values:
Difference in x-values = 2 - 1 = 1
4. Calculate the rate of change by dividing the difference in function values by the difference in x-values:
Rate of change = Difference in function values / Difference in x-values
= 10 / 1 = 10
Therefore, the rate of change between x = 1 and x = 2 for the function f(x) = x^3 + 3x is 10.
To find the equation of a secant line containing two points, we can use the point-slope form of a linear equation:
1. Plug the x and y values of the two given points into the point-slope form: y - y1 = m(x - x1)
Point 1: (1, f(1)) = (1, 4)
Point 2: (2, f(2)) = (2, 14)
2. Calculate the slope (m):
Slope (m) = (f(2) - f(1)) / (2 - 1) = (14 - 4) / (2 - 1) = 10
3. Choose one of the two points to plug into the point-slope equation. Let's use Point 1 (1, 4):
y - 4 = 10(x - 1)
4. Simplify and rearrange the equation to the standard form (Ax + By = C):
Distribute: y - 4 = 10x - 10
Combine like terms: y = 10x - 6
Therefore, the equation of the secant line containing the points (1, f(1)) and (2, f(2)) is y = 10x - 6.