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Essential Concepts
Integrals, Exponential Functions, and Logarithms
- The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. This section develops the concepts in a mathematically rigorous way.
- The cornerstone of the development is the definition of the natural logarithm in terms of an integral.
- The function ex is then defined as the inverse of the natural logarithm.
- General exponential functions are defined in terms of ex, and the corresponding inverse functions are general logarithms.
- Familiar properties of logarithms and exponents still hold in this more rigorous context.
Exponential Growth and Decay
- Exponential growth and exponential decay are two of the most common applications of exponential functions.
- Systems that exhibit exponential growth follow a model of the form y=y0ekt.
- In exponential growth, the rate of growth is proportional to the quantity present. In other words, y′=ky.
- Systems that exhibit exponential growth have a constant doubling time, which is given by (ln2)/k.
- Systems that exhibit exponential decay follow a model of the form y=y0e−kt.
- Systems that exhibit exponential decay have a constant half-life, which is given by (ln2)/k.
Calculus of the Hyperbolic Functions
- Hyperbolic functions are defined in terms of exponential functions.
- Term-by-term differentiation yields differentiation formulas for the hyperbolic functions. These differentiation formulas give rise, in turn, to integration formulas.
- With appropriate range restrictions, the hyperbolic functions all have inverses.
- Implicit differentiation yields differentiation formulas for the inverse hyperbolic functions, which in turn give rise to integration formulas.
- The most common physical applications of hyperbolic functions are calculations involving catenaries.
Key Equations
- Natural logarithm function
- lnx=∫x11tdt Z
- Exponential functiony=ex
- lny=ln(ex)=x Z
Glossary
- catenary
- a curve in the shape of the function y=acosh(x/a) is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary
- doubling time
- if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by (ln2)k
- exponential decay
- systems that exhibit exponential decay follow a model of the form y=y0e−kt
- exponential growth
- systems that exhibit exponential growth follow a model of the form y=y0ekt
- half-life
- if a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by (ln2)k
Study Tips
The Natural Logarithm as an Integral
- Visualize the logarithm as an area under a curve
- Practice using logarithm properties to manipulate expressions
- Remember the domain restriction (x>0) for lnx
Properties of the Exponential Function
- Visualize e as the x-coordinate where area under 1t curve equals 1
- Practice using exponential properties to manipulate expressions
- Connect differentiation and integration of exponential functions
General Logarithmic and Exponential Functions
- Practice converting between general exponential and logarithmic forms.
- Memorize the key derivative and integral formulas for ax and logax.
- Remember that all logarithmic functions are constant multiples of each other.
- When differentiating or integrating expressions with general bases, convert to natural logarithms or exponentials first if needed.
- Pay attention to the chain rule when dealing with composite functions involving exponentials or logarithms.
Exponential Growth Model
- Understand the significance of the growth constant k in determining the rate of growth.
- When solving exponential growth problems, use logarithms to isolate the variable of interest.
- For compound interest problems, pay attention to whether the interest is compounded continuously or at discrete intervals.
- Remember that the doubling time is constant in exponential growth, regardless of the initial quantity.
Exponential Decay Model
- Understand the significance of the decay constant k in determining the rate of decay.
- When solving exponential decay problems, use logarithms to isolate the variable of interest.
- For Newton’s Law of Cooling problems, pay attention to the ambient temperature and its effect on the decay rate.
- Remember that the half-life is constant in exponential decay, regardless of the initial quantity.
Derivatives and Integrals of the Hyperbolic Functions
- Memorize the definitions of sinh and cosh in terms of exponentials.
- Practice deriving the derivatives of hyperbolic functions from their definitions.
- Notice the patterns in the derivative formulas and compare them with trigonometric function derivatives.
- When integrating, look for opportunities to use u-substitution, especially with composites of hyperbolic functions.
- Remember that coshx is always positive, which can simplify some integrals (e.g., when dealing with absolute value).
- When solving problems, consider whether using hyperbolic function identities might simplify the expression.
Calculus of Inverse Hyperbolic Functions
- Memorize the domains and ranges of inverse hyperbolic functions.
- Notice the similarities between derivatives of inverse hyperbolic and inverse trigonometric functions.
- When integrating, pay attention to the form of the integrand to identify which inverse hyperbolic function to use.
- Remember that tanh−1x and coth−1x have the same derivative, so context is important when integrating 11−x2.
- When solving problems, consider whether using hyperbolic function identities might simplify the expression.
Applications of Hyperbolic Functions
- Understand the physical meaning of the parameters in the catenary equation y=acosh(xa).
- Practice sketching catenary curves for different values of ‘a‘ to understand how it affects the shape.
- When solving catenary problems, pay attention to the units of measurement given and ensure your answer uses the correct units.
- Practice deriving the derivative of catenary functions, as this is often needed in arc length calculations.