Show a numerical method for approximating the instantaneous rate of change at x = 3 for the function given by ƒ(x) = - 2x2 + 4x + 1 using slopes of secants.
find f(4), f(3.5), f(3.2), f(3.1), f(3.01)
Now find the slopes of the secants from (3,f(3)) to each of the points.
Watch how it converges to the tangent line at x=3.
To approximate the instantaneous rate of change at a specific point using slopes of secants, you can apply the concept of the difference quotient.
The difference quotient measures the average rate of change over a small interval. By taking smaller and smaller intervals, we can approach the instantaneous rate of change at a specific point.
In this case, we want to find the instantaneous rate of change of the function ƒ(x) = -2x^2 + 4x + 1 at x = 3. Let's use secants to estimate the derivative, or the instantaneous rate of change.
Step 1: Choose a small interval around x = 3
Let's pick a small interval, such as h = 0.1, centered at x = 3. The interval will go from x = 3 - h to x = 3 + h, which is x = 2.9 to x = 3.1.
Step 2: Calculate the slope of the secant line
To find the slope of the secant line passing through the points (x, ƒ(x)) and (x+h, ƒ(x+h)), where x = 3 and h = 0.1, we can use the formula for slope:
slope = (ƒ(x+h) - ƒ(x)) / (x+h - x)
Substituting the values into the formula, we have:
slope = (ƒ(3+0.1) - ƒ(3)) / (3+0.1 - 3)
Calculate ƒ(3+0.1):
ƒ(3+0.1) = -2(3+0.1)^2 + 4(3+0.1) + 1
ƒ(3+0.1) = -2(9.01) + 4(3.1) + 1
Calculate ƒ(3):
ƒ(3) = -2(3)^2 + 4(3) + 1
ƒ(3) = -18 + 12 + 1
Substituting the values into the slope formula, we have:
slope = (-18 + 12 + 1) / (0.1)
Step 3: Calculate the approximate instantaneous rate of change
Calculating the slope, we have:
slope = -5 / 0.1
This gives us the average rate of change over the interval from x = 2.9 to x = 3.1.
Step 4: Refine the estimate
To get a better approximation of the instantaneous rate of change at x = 3, we can repeat steps 1-3 with smaller and smaller intervals, approaching a difference of zero.
By taking smaller intervals, the average rate of change will converge to the instantaneous rate of change at x = 3.
Using the difference quotient and slopes of secants, we can approximate the instantaneous rate of change at x = 3 for the given function ƒ(x) = -2x^2 + 4x + 1.