5. Find the slope of the secant to the curve f(x) = 0.25(4)^x - 2 between:
i) x = 0 and x = 1
ii) x = 0 and x = 0.5
iii) x = 0 and x = 0.1
iv) x = 0 and x = 0.01
is it
f(x) = .25(4^x) - 2 , which is what it would be according to your typing
or , is it
f(x) = .25(4)^(x-2) , I think that is what you wanted
let me know before I work at it.
It's f(x) = .25(4^x) - 2
Thank you!
To find the slope of the secant to the curve, we need to use the formula for slope:
slope = (f(x2) - f(x1))/(x2 - x1)
Where f(x) is the given function and x1 and x2 are the two points on the curve between which we want to find the slope.
Let's calculate the slope for each pair of points:
i) x = 0 and x = 1
Here, x1 = 0 and x2 = 1.
We need to find f(x1) and f(x2) to substitute them into the slope formula.
f(x1) = 0.25 * (4)^0 - 2 = 0.25 * 1 - 2 = -1.75
f(x2) = 0.25 * (4)^1 - 2 = 0.25 * 4 - 2 = 0.5
Now we can substitute the values into the slope formula:
slope = (0.5 - (-1.75))/(1 - 0)
slope = 2.25/1
slope = 2.25
ii) x = 0 and x = 0.5
Here, x1 = 0 and x2 = 0.5.
f(x1) = 0.25 * (4)^0 - 2 = 0.25 * 1 - 2 = -1.75
f(x2) = 0.25 * (4)^0.5 - 2
We need to calculate f(x2) before we can substitute the values into the slope formula.
f(x2) = 0.25 * √4 - 2 = 0.25 * 2 - 2 = -1.5
Now we can substitute the values into the slope formula:
slope = (-1.5 - (-1.75))/(0.5 - 0)
slope = 0.25/0.5
slope = 0.5
iii) x = 0 and x = 0.1
Here, x1 = 0 and x2 = 0.1.
f(x1) = 0.25 * (4)^0 - 2 = 0.25 * 1 - 2 = -1.75
f(x2) = 0.25 * (4)^0.1 - 2
We need to calculate f(x2) before we can substitute the values into the slope formula.
Now we can substitute the values into the slope formula:
slope = (f(x2) - f(x1))/(x2 - x1)
slope = (1.0625 - (-1.75))/(0.1 - 0)
slope = 2.8125/0.1
slope = 28.125
iv) x = 0 and x = 0.01
Here, x1 = 0 and x2 = 0.01.
f(x1) = 0.25 * (4)^0 - 2 = 0.25 * 1 - 2 = -1.75
f(x2) = 0.25 * (4)^0.01 - 2
We need to calculate f(x2) before we can substitute the values into the slope formula.
Now we can substitute the values into the slope formula:
slope = (f(x2) - f(x1))/(x2 - x1)
slope = (0.684 - (-1.75))/(0.01 - 0)
slope = 2.434/0.01
slope = 243.4
Therefore, for the given function f(x) = 0.25(4)^x - 2, the slopes of the secants between the specified points are:
i) The slope between x = 0 and x = 1 is 2.25.
ii) The slope between x = 0 and x = 0.5 is 0.5.
iii) The slope between x = 0 and x = 0.1 is 28.125.
iv) The slope between x = 0 and x = 0.01 is 243.4.