2. You can identify where a function increases and decreases by finding the derivative. When you find the derivative, you can plug in a value for x, in order to find the slope of the function at that point.
3. The Chain Rule is a method of find the derivative of a composite function. you begin by working from the outside of the function in. First, you take the derivative of the outside function. Second, you rewrite the interior function. Finally, you multiply by the derivative of the interior function.
f'(x) = sqrt(1 -2x)f'(x) = (1 - 2x)^(1/2)
f'(x) = 1/2(1 - 2x) ^ (-1/2) (-2)
f'(x) = -2/2(1 - 2x)^(1/2)
f'(x) = -1/sqrt(1 - 2x)
Find tangent line at x = -7.5
f(-7.5) = sqrt(1 -2(-7.5)) = sqrt(16)
f(-7.5) = 4
f'(-7.5) = -1 / sqrt ( 1 -2(-7.5))
f'(-7.5) = -1/4
tangent line = y - 4 = -1/4 (x + 7.5)
4. h(x) = f(g(x)) g(-4) = 5 g'(-4) =2 f'(5) = 20. Find h'(-4)
h'(-4) = f'(g(-4))(2)
h'(-4) = f'(5)(2)
h'(-4) =(20)(2)
h'(-4) = 40
I like how you used an example of a square root. This is something that myself and some others struggle with, especially since it deals with fractions and negative numbers as exponents.
ReplyDeleteI like your organization strategy. It clearly lays out progression of solving the problem and makes your work easy to follow. Thank you for being such a wonderful patron of mathematics.
ReplyDeleteCheck your defn in the first part. Does y = x^3 change direction at x =0?
ReplyDelete