Reflection

To begin, this week in Calculus we continued learning section 2.2 and we also began learning section 2.3. In section 2.2 we learned about a graphical approach of limits, how to find the average velocity, finding acceleration, and finding velocity. In the section 2.3, which we started learning recently, we began practicing problems about the product rule and the quotient rule.

Section 2.2
Example:
Find the average rate of change of the function over the given interval.
f(t) = 4t +5 [1, 2]
(1, 9) = 4(1) + 5 = 9
(2, 13) = 4(2) + 5 = 13

m = y2 – y1 / x2 – x1
= 13 – 9 / 2 -1
= 4/1
=4

Section 2.3
Product Rule:
f(x)g(x) = f(x)g’(x) + g(x)f’(x)
(recopy 1st)(derivative 2nd) + (recopy 2nd)(derivative 1st)
Example: g(x) = (x^2 + 3)(x^2 – 4x)
g’(x) = (x^2 + 3)(2x – 4) + (x^2 – 4x)(2x)
= 2x^3 – 12 +2x^3 – 8x^2
= 4x^3 – 8x^2 – 12 / 4
= x^3 -2x^2 -3
Quotient Rule:
f(x)/g(x) = g(x)f’(x) – [f(x)g’(x)] / (g(x))^2
(recopy bottom)(derivative top) – [(recopy top)(derivative bottom)] / (bottom)^2
Example: f(x) = x / x^2 + 1
f’(x) = (x^2 + 1)(1) – [(x)(2x)] / (x^2+1)^2
= x^2 + 1 – 2x^2 / (x^2 + 1)^2
= -x^2 + 1 / (x^2 + 1)^2