Use a linear approximation (or differentials) to estimate the given number.
e0.01
y = e^x
y' = e^x
at x=1, y=1
The line through (0,1) with slope=1 is
y = x+1
so, e^0.01 ~ 1+0.01 = 1.01
Note that e^x = 1 + x + 1/2 x^2 + ...
so as a line, e^x ~ 1+x near x=0.
Why did the number go to therapy? It had an e-xistential crisis and needed some exponential healing!
But speaking of estimating e0.01 using a linear approximation, let's get down to business.
Using a linear approximation, we can approximate the value of e0.01 by considering the tangent line to the graph of the exponential function e^x at x = 0.
The equation for this tangent line is y = e^0 + (e^0)(x - 0), which simplifies to y = 1 + x.
Now, if we plug in x = 0.01 into this linear approximation equation, we get y ≈ 1 + 0.01 = 1.01.
So, using a linear approximation, we estimate that e0.01 is approximately 1.01. Keep in mind that this is just an approximation, so don't take it too seriously!
To estimate the number e^0.01 using linear approximation or differentials, we can use the derivative of the function e^x.
The derivative of e^x with respect to x is simply e^x. So, we have f'(x) = e^x.
Now, using the linear approximation formula, we can approximate a small change in the function value by multiplying the derivative with the change in the input variable. In this case, the change in the input variable is 0.01 (Δx), and we want to find the approximation of e^0.01.
So, the linear approximation can be written as:
Δf = f'(x) * Δx
Substituting the values, we get:
Δf = e^0.01 * 0.01
To find the approximation of e^0.01, we need to add this change in the function value to the original value at x = 0:
Approximation of e^0.01 = e^0 + Δf = e^0 + e^0.01 * 0.01
Since e^0 equals 1, we can simplify this to:
Approximation of e^0.01 = 1 + e^0.01 * 0.01
Calculating this expression will give you an estimated value for e^0.01 using linear approximation or differentials.
Kela, 8 consecutive questions without any indication on your part to show us where your problem is.
Please show us some effort and steps so we can see where you are having problems.
As it stands it looks like a typical case of "homework dumping"