given the functionf(x)=x^3+3x find the rateof change between the two stated values for x:1 to 2
find the equation of a secant line containig the given points (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)
To find the rate of change between two values of x, we need to calculate the slope of the secant line that connects the two points.
First, let's find the values of f(1) and f(2) by substituting the values of x into the given function:
f(1) = (1)^3 + 3(1) = 1 + 3 = 4
f(2) = (2)^3 + 3(2) = 8 + 6 = 14
Now we have the coordinates of the two points: (1, 4) and (2, 14). We can use the slope formula to find the slope of the secant line:
slope = (change in y) / (change in x)
= (f(2) - f(1)) / (2 - 1)
= (14 - 4) / 1
= 10
Therefore, the rate of change (slope) between the two given values of x is 10.
Now, let's find the equation of the secant line containing the points (1, f(1)) and (2, f(2)). We already have the slope, so we can use the point-slope form:
y - y1 = m(x - x1)
Using the coordinates of the first point (1, 4), we get:
y - 4 = 10(x - 1)
Simplifying,
y - 4 = 10x - 10
To get the equation in slope-intercept form (y = mx + b), we can rearrange:
y = 10x - 10 + 4
y = 10x - 6
Therefore, the equation of the secant line containing the points (1, f(1)) and (2, f(2)) is y = 10x - 6.