University:
York University
Course:
MATH 1013 | Applied Calculus I
Academic year:

2022
Views:

464

Pages:

3
Author:

Nick T.

3.1 Power Functions Monomial: y=cx n n=0→y=c (constant function) n=1→y=x (linear function) Power Rule: dx d x n =nx n−1 (applies to n any real number) Examples: y=x 3 −3x 2 − x 3 y= 3 x 2 =x 3 2 y=x 5 4 y=x 3π =3πx 3π−1 y=xe x =e x +xe x−1 y= x 2 3 5 −3x 4 5 +7 x − x 3 x 3 +2x 2 +7 5x 2 −3 −3x 4 5 +7x 2 1 −x −1 +2x −3 +7 2 −15 x 2 −5 − 4 15 x 4 1 + 2 7 x 2 −1 +x −2 + x 4 6 lim(h→0) (b^(x+h) - b^x) / h = lim(h→0) b^x(b^h - 1) / h b^x lim(h→0) (b^h - 1) / h = constant f(x) = b^x f(0) = b^0 = 1 f'(0) = lim(h→0) (f(0+h) - f(0)) / h = lim(h→0) (b^h - 1) / h f'(x) = b^x * f'(0) f'(x) = b^x * ln(b) Examples: y = 3^x → y' = 3^x ln(3) y = 5^x → y' = 5^x ln(5) d/dx [2^x] = 0.693147 lim(h→0) (b^h - 1) / h = b^x * f'(0) f'(1) = e^1 = e f and f' f(x) = e^x f'(x) = e^x f'(1) = e^1 = e f'(x) = e^x f'(x) = e^x Chapter 3.2 y = f(x)g(x) f(x+h)g(x+h) - f(x)g(x) ------------------------- h lim(h→0) [g(x+h)b(x+h) - g(x)b(x)] / h lim(h→0) [g(x)b(x+h) + g(x+h)b(x) + g(x)b(x+h) + g(x+h)b(x) - g(x)b(x)] / h lim(h→0) [g(x) Δb(x) + b(x) Δg(x) + Δg(x) Δb(x)] / h g(x) db/dx + b(x) dg/dx + 0 f(x) = g(x)b(x) → f'(x) = g'(x)b(x) + g(x)b'(x) y = f(x)g(x) → y' = f'(x)g(x) + f(x)g'(x) - Product rule.