BackBusiness Calculus: Core Algebra and Applications Study Guide
Study Guide - Smart Notes
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Factoring Quadratics
Introduction
Factoring quadratics is a foundational algebraic skill, essential for solving quadratic equations and simplifying expressions in business calculus.
Factoring involves rewriting a quadratic expression as a product of two binomials.
Key steps include multiplying the leading coefficient by the constant, splitting the middle term, grouping, and factoring.
Example
Factor :
Multiply:
Numbers:
Rewrite:
Group:
Factor:
Final:
Linear Equations & Graphing
Introduction
Linear equations describe straight lines and are fundamental in modeling relationships in business and economics.
Find the slope using two points:
Use the point-slope form, then convert to slope-intercept form:
Example
Line through (0,3) and (4,7):
Slope:
Equation:
Final:
Quadratic Functions (Vertex Form)
Introduction
Quadratic functions in vertex form are useful for identifying the maximum or minimum point (vertex) of a parabola, which is often relevant in business optimization problems.
Vertex form:
Vertex:
Axis of symmetry:
Solve for x-intercepts if possible by setting
Example
Vertex:
Axis of symmetry:
Piecewise Functions
Introduction
Piecewise functions are defined by different expressions over different intervals and are used to model situations with abrupt changes, such as tax brackets or shipping rates.
Check which rule applies for a given x-value.
Use open/closed circles on graphs to indicate inclusion/exclusion of endpoints.
Rational Functions
Introduction
Rational functions are ratios of polynomials and are important in modeling rates, proportions, and marginal analysis in business calculus.
Vertical asymptote: Set denominator equal to zero and solve for x.
Horizontal asymptote: Compare the degrees of the numerator and denominator.
Example
Vertical asymptotes:
Horizontal asymptote:
Difference Quotient
Introduction
The difference quotient is a fundamental concept in calculus, representing the average rate of change of a function over an interval. It is the basis for the derivative.
Formula:
Expand , subtract , and divide by .
Example
For :
Subtract :
Divide by :
Exponential & Logarithmic Functions
Introduction
Exponential and logarithmic functions are used to model growth and decay, such as compound interest, population growth, and depreciation.
Exponential:
Domain:
Range:
Logarithmic:
Domain:
Range:
Example
For :
Domain:
Range:
Supply & Demand
Introduction
Supply and demand equations are used in economics to determine equilibrium price and quantity in a market.
Set supply and demand equations equal to each other to find equilibrium.
Solve for the variable (usually quantity or price).
Plug back to find the corresponding value.
Example
Supply: Demand:
Set equal:
Solve:
Equilibrium: 150 units,
Financial Applications
Introduction
Financial mathematics in business calculus includes compound interest and loan payment calculations, which are essential for business decision-making.
Compound interest formula:
Loan payment formula (PMT):
Example A: Compound Interest
annual, compounded quarterly, 3 years:
Example B: Loan Payment
annual, 2 years, monthly payments:
