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

2023
Views:

340

Pages:

2
Author:

Niklas

II. Exponential Functions (Ch. 3-1) I. DEFINITION exponential function: function with the exponent being variable (f(x)=ab^{x}) note (a\ne0, b 0, b\ne1) if (a=0, f(x)=0) if (b 0, f(x)) will oscillate if (b=1, f(x)=a) II. GROWTH AND DECAY (only defined for (a 0)) exponential growth: ex: COVID-19 spread condition: (b 1) or coeff of x is pos graph: [graph of exponential growth] y-int: 1 domain: ((-∞,∞)) range: ((0,∞)) HA: (y=0) exponential decay: ex: radioactive decay condition: (0 1) or coeff of x is neg graph: [graph of exponential decay] y-int: 1 domain: ((-∞,∞)) range: ((0,∞)) HA: (y=0) III. SOLVING EXPONENTIAL EQUATIONS (b^{m}=b^{n}\Rightarrow m=n) Examples: (10^{x}=1000) (100^{x\cdot2}=1000^{x}) ((\frac{1}{5})^{x}=125) (4^{2x+3}=1) (10^{x}=\frac{1}{10}) (81\cdot9^{-2x-2}=27) (2^{2x+4}=9^{4x}) IV. GRAPHING EXPONENTIAL FUNCTIONS: (f(x)=ab^{cx}+d) domain: ((-∞,∞)) HA: (y=d) continuous on: ((-∞,∞)) inc./dec.: ((-∞,∞)) y-int: (ad) if (a 0), the graph will reflect over the x-axis (a) and (c) can dilate the graph (d) affects the vertical shift Graph of (f(x)=3^{x}): domain: ((-∞,∞)) range: ((0,∞)) y-int: 1 HA: (y=0) inc.: ((-∞,∞)) dec.: none end behavior: as (x→∞, y→∞) as (x→-∞, y→0) Graph of (f(x)=2^{x}-1): domain: ((-∞,∞)) range: ((-1,∞)) y-int: 0 HA: (y=-1) inc.: ((-∞,∞)) dec.: none end behavior: as (x→∞, y→∞) as (x→-∞, y→-1) Graph of (f(x)=-\frac{1}{2}(2^{x})+4): domain: ((-∞,∞)) range: ((-∞,4)) y-int: 3.5 HA: (y=4) inc.: none dec.: ((-∞,∞)) end behavior: as (x→∞, y→-∞) as (x→-∞, y→4)