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

2022
Views:

311

Pages:

4
Author:

Nick T.

Calculus Subheadings: Functions Symmetry Trigonometric Functions Inverse Trigonometric Functions Text: Domain Range Vertical Line Test Symmetry Odd Functions Even Functions Trigonometric Functions Inverse Trigonometric Functions Domain x (trig ratio) Range y (angle) Diagrams: A right triangle with sides labeled a, b, and c Graphs of various functions, including: Polynomial functions Trigonometric functions (sine, cosine, tangent, secant, cosecant, cotangent) Inverse trigonometric functions (arcsine, arccosine, arctangent) Intra to logarithms Log(16)-x Loga(16)-4 Log(1/6) = 2^x - 16 2^0 = 1 2^-3 = 1/8 Middle Section (Graphs): Three graphs are plotted on the same coordinate plane: y = 2cos(x) y = cos(x) y = cos(2x) y = 1/2 cos(x) y = cos(1/2x) Bottom Section: Transforming functions y = f(x) + 6 (shift upwards) y = f(x - t) (shift to the right) Stretching and Reflecting y = -f(x) (reflection across the x-axis) y = f(-x) (reflection across the y-axis) y = kf(x) (stretch vertically by a factor of k) y = f(kx) (shrink horizontally by a factor of k) Exponents a^m * a^n = a^(m+n) (a^m)^n = a^(mn) a^-n = 1/a^n (a/b)^-n = (b/a)^n a^0 = 1 a^1 = a a^2 = a * a a^3 = a * a * a a^4 = a * a * a * a a^5 = a * a * a * a * a a^6 = a * a * a * a * a * a a^7 = a * a * a * a * a * a * a a^8 = a * a * a * a * a * a * a * a The Number e e = 2.71828... Natural exponential function lim(1+1/n)^n = e ln(e^x) = x Laws of logarithms log_b(xy) = log_b(x) + log_b(y) log_b(x/y) = log_b(x) - log_b(y) log_b(x^r) = r log_b(x) (where r is any real number) Logarithms log_e(67) = ln(67) log_b(x) = y <=> b^y = x Inverse Functions N = f(t) t = f^-1(N) One-to-one functions Horizontal line test f(y) = x <=> f^-1(x) = y domain of f = range of f^-1 range of f = domain of f^-1 Tangent line slope f'(1) = 2 y - f(a) = f'(a)(x - a) m = lim(h->0) [f(a+h) - f(a)]/h Middle Section (Graphs): Graphs of logarithmic functions with different bases (b = 2, e, 10) Graph of the natural logarithm (ln(x))