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Mathematical Concepts & Applications: Exponential, Logarithmic, and Trigonometric Functions

Explore exponential functions, logarithmic equations, compound interest, trigonometric identities, and more in this comprehensive guide by Donna Ball and Pam. Dive into graphing basics, exponentials, logarithms, and solving equations with practical examples. Understand growth rates, decay, and compound interest formulas for real-world applications.

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Mathematical Concepts & Applications: Exponential, Logarithmic, and Trigonometric Functions

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  1. MATH! by: Donna Ball and Pam

  2. 5.2 Exponential Functions & Graphs • F(x)=ax • x= real # • a>0, a 1 • Graphing Basics • Base e: • f(x)=ex, g(x)=e-x • Compound Interest: • A=P(1+ (r/n))nt • P=initial value, r=rate, n=amount compounded annually, t=time

  3. Ch. 5.2 Example

  4. 5.3 Logarithmic Functions & Graphs • Log Function Equation: • y=logax • x>0 • a=positive #, a 1 • General Rules: • loga1=0, ln1=0 • logaa=1, lne=1 • Log to Exponential: • logax=yx=ay • Change of Base: • logbM=(logaM/logab)

  5. Ch. 5.3 Example

  6. 5.4 Properties of Logarithmic Functions • Product Rule: • logaMN=logaM+logaN • Power Rule: • logaMp=plogaM • Quotient Rule: • loga(M/N)=logaM-logaN • Logarithm of a Base to a Power: • logaax=x • Base to a Logarthimic Power: • Alogax=x

  7. Ch. 5.4 Example

  8. 5.5 Solving Exponential & Logarithmic Equations • Base-Exponent Property: • ax=ayx=y • a>0, a (can't)=1 • Property of Logarithmic Equality: • logaM=logaNM=N • M>0, N>0, a>0, a (can't)=1

  9. Ch. 5.5 Example

  10. 5.6 Growth, Decay, & Compound Interest • Growth Equation: • P(t)=Poekt • k>0 • Growth Rate & Doubling Time: • KT=ln2 • K=(ln2/T) • T=(ln2/K) • Exponential Decay: • P(t)=Poe-kt • k>0 • Decay Rate & Half Life: • KT=ln2 • K=(ln2/T) • T=(ln2/K)

  11. Ch. 5.6 Example

  12. Ch. 5.6 Example (continued)

  13. 7.1 Pythagorean and Sum and Difference • Basic Identities: • Pythagorean Identities: • Sum & Difference Identities:

  14. Ch. 7.1 Example

  15. 7.2 Cofunctions, Double-Angle, & Half-Angle • Cofunction Identities: • Double-Angle Identities: • Half-Angle Identities:

  16. Ch. 7.2 Example (cofunctions)

  17. 7.3 Proving Trigonometric Identities • Method 1: • Start with one side and solve for opposite side. • Method 2: • Solve both sides until they're equal to each other. • Product-to-Sum Identities: • Sum-to-Product Identities:

  18. Ch. 7.3 Example

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