y = g(x) = cos(x)
Can someone show how to estimate g'(pi/2) using the limit definition of the derivative and different values of h.
Thanks!
g'(x) ≈ ∆g/∆x = [g(x+h)-g(x)]/h
So, just pick a small h, and figure
[g(π/2 + h)-g(π/2)]/h
= [cos(π/2 + h)-cos(π/2)]/h
= [(cos(π/2)cos(h) - sin(π/2)sin(h)-cos(π/2)]/h
= -sin(h)/h
So, pick any value of h close to zero, and that approximates g'(π/2). g' is exactly the limit as h->0.
Thank You very much, I understand it now
Certainly! To estimate g'(pi/2) using the limit definition of the derivative, we'll need to find the value of the derivative at x = pi/2 by taking the limit as h approaches 0. Here's how you can do it step by step:
1. Start with the function g(x) = cos(x). We want to find g'(pi/2), which represents the slope of the graph at x = pi/2.
2. Recall that the limit definition of the derivative is given by:
g'(x) = lim(h -> 0) [g(x + h) - g(x)] / h
3. Rewrite the formula using x = pi/2:
g'(pi/2) = lim(h -> 0) [g(pi/2 + h) - g(pi/2)] / h
4. Substitute in the function g(x) = cos(x):
g'(pi/2) = lim(h -> 0) [cos(pi/2 + h) - cos(pi/2)] / h
5. Simplify the expression inside the limit:
g'(pi/2) = lim(h -> 0) [cos(pi/2 + h) - 0] / h
Since cos(pi/2) = 0.
6. Now, let's evaluate this expression for different values of h, approaching 0. Choose a few values of h that are very close to 0, such as h = 0.1, 0.01, 0.001, etc.
For h = 0.1, substitute it into the expression:
g'(pi/2) ≈ [cos(pi/2 + 0.1) - 0] / 0.1
Calculate the value using a calculator or mathematical software.
7. Repeat the process for the other chosen values of h, such as h = 0.01, 0.001, etc. each time plugging them into the expression and calculating the result.
8. As you use smaller values of h, you'll notice that the values you obtain become closer to a certain value. This value will be your estimation of g'(pi/2) using the limit definition of the derivative.
Note: Keep in mind that the more values of h you try, the more accurate your estimation will be. To get a precise estimate, you can use a computer program or mathematical software that can handle very small values of h and perform the calculations for you.
I hope this helps you estimate g'(pi/2) using the limit definition of the derivative!