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Integration of Exponential, Logarithmic, and Hyperbolic Functions: Cheat Sheet

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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=y0ekt.
  • 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=y0ekt
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 tanh1x and coth1x have the same derivative, so context is important when integrating 11x2.
  • 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.