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Essential Concepts

Limits at Infinity and Asymptotes

• If [latex]c[/latex] is a critical point of [latex]f[/latex] and [latex]f^{\prime}(x)>0[/latex] for [latex]xc[/latex], then [latex]f[/latex] has a local maximum at [latex]c[/latex].
• If [latex]c[/latex] is a critical point of [latex]f[/latex] and [latex]f^{\prime}(x)<0[/latex] for [latex]x0[/latex] for [latex]x>c[/latex], then [latex]f[/latex] has a local minimum at [latex]c[/latex].
• For a polynomial function [latex]p(x)=a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0[/latex], where [latex]a_n \ne 0[/latex], the end behavior is determined by the leading term [latex]a_n x^n[/latex]. If [latex]n\ne 0[/latex], [latex]p(x)[/latex] approaches [latex]\infty[/latex] or [latex]−\infty[/latex] at each end.
• For a rational function [latex]f(x)=\frac{p(x)}{q(x)}[/latex], the end behavior is determined by the relationship between the degree of [latex]p[/latex] and the degree of [latex]q[/latex]. If the degree of [latex]p[/latex] is less than the degree of [latex]q[/latex], the line [latex]y=0[/latex] is a horizontal asymptote for [latex]f[/latex]. If the degree of [latex]p[/latex] is equal to the degree of [latex]q[/latex], then the line [latex]y=\frac{a_n}{b_n}[/latex] is a horizontal asymptote, where [latex]a_n[/latex] and [latex]b_n[/latex] are the leading coefficients of [latex]p[/latex] and [latex]q[/latex], respectively. If the degree of [latex]p[/latex] is greater than the degree of [latex]q[/latex], then [latex]f[/latex] approaches [latex]\infty[/latex] or [latex]−\infty[/latex] at each end.

Applied Optimization Problems

• To solve an optimization problem, begin by drawing a picture and introducing variables.
• Find an equation relating the variables.
• Find a function of one variable to describe the quantity that is to be minimized or maximized.
• Look for critical points to locate local extrema.

L’Hôpital’s Rule

• L’Hôpital’s rule can be used to evaluate the limit of a quotient when the indeterminate form [latex]\frac{0}{0}[/latex] or [latex]\frac{\infty}{\infty}[/latex] arises.
• L’Hôpital’s rule can also be applied to other indeterminate forms if they can be rewritten in terms of a limit involving a quotient that has the indeterminate form [latex]\frac{0}{0}[/latex] or [latex]\frac{\infty}{\infty}[/latex].
• The exponential function [latex]e^x[/latex] grows faster than any power function [latex]x^p[/latex], [latex]p>0[/latex].
• The logarithmic function [latex]\ln x[/latex] grows more slowly than any power function [latex]x^p[/latex], [latex]p>0[/latex].

Newton’s Method

• Newton’s method approximates roots of [latex]f(x)=0[/latex] by starting with an initial approximation [latex]x_0[/latex], then uses tangent lines to the graph of [latex]f[/latex] to create a sequence of approximations [latex]x_1,x_2,x_3, \cdots[/latex].
• Typically, Newton’s method is an efficient method for finding a particular root. In certain cases, Newton’s method fails to work because the list of numbers [latex]x_0,x_1,x_2, \cdots[/latex] does not approach a finite value or it approaches a value other than the root sought.
• Any process in which a list of numbers [latex]x_0,x_1,x_2, \cdots[/latex] is generated by defining an initial number [latex]x_0[/latex] and defining the subsequent numbers by the equation [latex]x_n=F(x_{n-1})[/latex] for some function [latex]F[/latex] is an iterative process. Newton’s method is an example of an iterative process, where the function [latex]F(x)=x-\left[\frac{f(x)}{f^{\prime}(x)}\right][/latex] for a given function [latex]f[/latex].

Antiderivatives

• If [latex]F[/latex] is an antiderivative of [latex]f[/latex], then every antiderivative of [latex]f[/latex] is of the form [latex]F(x)+C[/latex] for some constant [latex]C[/latex].
• Solving the initial-value problem
  [latex]\frac{dy}{dx}=f(x),y(x_0)=y_0[/latex]

  requires us first to find the set of antiderivatives of [latex]f[/latex] and then to look for the particular antiderivative that also satisfies the initial condition.

Key Equations

• Infinite Limits from the Left
  [latex]\underset{x\to a^-}{\lim}f(x)=+\infty[/latex]
  [latex]\underset{x\to a^-}{\lim}f(x)=−\infty[/latex]
• Infinite Limits from the Right
  [latex]\underset{x\to a^+}{\lim}f(x)=+\infty[/latex]
  [latex]\underset{x\to a^+}{\lim}f(x)=−\infty[/latex]
• Two-Sided Infinite Limits
  [latex]\underset{x\to a}{\lim}f(x)=+\infty: \underset{x\to a^-}{\lim}f(x)=+\infty[/latex] and [latex]\underset{x\to a^+}{\lim}f(x)=+\infty[/latex]
  [latex]\underset{x\to a}{\lim}f(x)=−\infty: \underset{x\to a^-}{\lim}f(x)=−\infty[/latex] and [latex]\underset{x\to a^+}{\lim}f(x)=−\infty[/latex]

Glossary

antiderivative

a function [latex]F[/latex] such that [latex]F^{\prime}(x)=f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex] is an antiderivative of [latex]f[/latex]

end behavior

the behavior of a function as [latex]x\to \infty[/latex] and [latex]x\to −\infty[/latex]

horizontal asymptote

if [latex]\underset{x\to \infty }{\lim}f(x)=L[/latex] or [latex]\underset{x\to −\infty }{\lim}f(x)=L[/latex], then [latex]y=L[/latex] is a horizontal asymptote of [latex]f[/latex]

indeterminate forms

when evaluating a limit, the forms [latex]0/0[/latex], [latex]\infty / \infty[/latex], [latex]0 \cdot \infty[/latex], [latex]\infty -\infty[/latex], [latex]0^0[/latex], [latex]\infty^0[/latex], and [latex]1^{\infty}[/latex] are considered indeterminate because further analysis is required to determine whether the limit exists and, if so, what its value is

indefinite integral

the most general antiderivative of [latex]f(x)[/latex] is the indefinite integral of [latex]f[/latex]; we use the notation [latex]\displaystyle\int f(x) dx[/latex] to denote the indefinite integral of [latex]f[/latex]

infinite limit at infinity

a function that becomes arbitrarily large as [latex]x[/latex] becomes large

initial value problem

a problem that requires finding a function [latex]y[/latex] that satisfies the differential equation [latex]\frac{dy}{dx}=f(x)[/latex] together with the initial condition [latex]y(x_0)=y_0[/latex]

iterative process

process in which a list of numbers [latex]x_0,x_1,x_2,x_3, \cdots[/latex] is generated by starting with a number [latex]x_0[/latex] and defining [latex]x_n=F(x_{n-1})[/latex] for [latex]n \ge 1[/latex]

L’Hôpital’s rule

if [latex]f[/latex] and [latex]g[/latex] are differentiable functions over an interval [latex]a[/latex], except possibly at [latex]a[/latex], and [latex]\underset{x\to a}{\lim} f(x)=0=\underset{x\to a}{\lim} g(x)[/latex] or [latex]\underset{x\to a}{\lim} f(x)[/latex] and [latex]\underset{x\to a}{\lim} g(x)[/latex] are infinite, then [latex]\underset{x\to a}{\lim}\dfrac{f(x)}{g(x)}=\underset{x\to a}{\lim}\dfrac{f^{\prime}(x)}{g^{\prime}(x)}[/latex], assuming the limit on the right exists or is [latex]\infty[/latex] or [latex]−\infty[/latex]

limit at infinity

the limiting value, if it exists, of a function as [latex]x\to \infty[/latex] or [latex]x\to −\infty[/latex]

Newton’s method

method for approximating roots of [latex]f(x)=0[/latex]; using an initial guess [latex]x_0[/latex], each subsequent approximation is defined by the equation [latex]x_n=x_{n-1}-\dfrac{f(x_{n-1})}{f^{\prime}(x_{n-1})}[/latex]

oblique asymptote

the line [latex]y=mx+b[/latex] if [latex]f(x)[/latex] approaches it as [latex]x\to \infty[/latex] or [latex]x\to −\infty[/latex]

optimization problems

problems that are solved by finding the maximum or minimum value of a function

Study Tips

Limits at Infinity

• Visualize the behavior of functions as [latex]x[/latex] approaches infinity.
• Connect limits at infinity to the concept of horizontal asymptotes.
• Understand the difference between a finite limit and an infinite limit at infinity.
• Practice applying formal definitions to prove limits at infinity.

End Behavior

• Practice identifying end behavior for various function types.
• For polynomials, focus on the highest degree term’s sign and parity.
• For rational functions, compare degrees of numerator and denominator.
• Remember that functions can cross horizontal asymptotes.
• Memorize the end behavior of basic transcendental functions.

Drawing Graphs of Functions

• Use a systematic approach for every function, even if some steps seem unnecessary.
• Sketch the graph as you go through each step, refining it with new information.
• Pay special attention to behavior near critical points and asymptotes.
• Confirm your analysis by using graphing technology.

Solving Optimization Problems

• Always start by clearly defining variables and what they represent.
• Sketch the problem scenario when possible to visualize constraints.
• Use the problem constraints to express the objective function in terms of a single variable.
• For closed intervals, remember to check the endpoints as well as critical points.
• For unbounded intervals, analyze the behavior of the function as variables approach infinity or zero.
• When dealing with geometric problems, recall relevant formulas (area, volume, Pythagorean theorem, etc.).
• After finding a solution, always check if it makes sense in the context of the problem.
• Practice identifying the domain of functions to determine potential intervals for optimization.
• For rational functions, focus on points where the denominator could be zero.
• Remember that the absolute extrema might occur at boundary points of the domain, not just at critical points.

L’Hôpital’s Rule

• Identify indeterminate forms before applying L’Hôpital’s Rule.
• Practice recognizing limits that lead to [latex]\frac{0}{0}[/latex] or [latex]\frac{\infty}{\infty}[/latex] forms.
• Remember that L’Hôpital’s Rule involves taking derivatives of numerator and denominator separately.
• Be prepared to apply the rule multiple times if necessary.
• Always check if the limit after applying L’Hôpital’s Rule is still indeterminate.
• For rational functions, compare degrees of numerator and denominator as an alternative method.
• Be cautious with trigonometric functions near zero, as they often lead to indeterminate forms.
• Remember that L’Hôpital’s Rule is not the only method for evaluating limits. Consider other techniques when appropriate.

Other Indeterminate Forms

• For [latex]0 \cdot \infty[/latex] forms, try rewriting as [latex]\frac{\text{term approaching 0}}{\frac{1}{\text{term approaching }\infty}}[/latex].
• For [latex]\infty - \infty[/latex] forms, look for a common denominator to combine terms.
• For exponential indeterminate forms, use the natural log to rewrite the expression.
• Be prepared to apply L’Hôpital’s Rule multiple times if necessary.
• When dealing with exponential forms, consider both [latex]\frac{\ln(\text{base})}{\frac{1}{\text{exponent}}}[/latex] and [latex]\frac{\text{exponent}}{\frac{1}{\ln(\text{base})}}[/latex] as possible rewrite options.
• Practice algebraic manipulation to rewrite expressions in forms suitable for L’Hôpital’s Rule.
• Remember that these indeterminate forms don’t have a single, predictable behavior – analysis is always necessary.
• Watch for opportunities to simplify expressions before applying L’Hôpital’s Rule.

Growth Rates of Functions

• When comparing growth rates, always consider the behavior as [latex]x \to \infty[/latex].
• Use L’Hôpital’s Rule to evaluate limits when comparing growth rates.
• Remember that exponential functions generally grow faster than power functions.
• Power functions grow faster than logarithmic functions.
• Create tables of values to visualize growth rates for different functions.
• Graph functions together to visually compare their growth rates.
• Pay attention to the base of exponential functions when comparing growth rates.
• Consider the exponent of power functions when comparing their growth rates.

Approximating with Newton’s Method

• Always check if the derivative [latex]f'(x)[/latex] is defined and non-zero at each iteration.
• Be aware that Newton’s Method may converge to a different root than intended.
• Be prepared to perform multiple iterations to achieve desired accuracy.
• Watch for signs of failure, such as repeating or diverging values.

Finding the Antiderivative

• Remember to always include the constant of integration [latex]C[/latex] when expressing the general antiderivative.
• Pay attention to the domain of the function when finding antiderivatives, especially for functions like [latex]1/x[/latex].
• Practice verifying your antiderivative by differentiating it.
• Look for patterns in antiderivatives that correspond to derivative rules (e.g., power rule, exponential rule).
• Consider the geometric interpretation: antiderivatives of a function are the family of curves whose slopes are given by the function.

Indefinite Integrals

• Practice recognizing basic integral forms and their antiderivatives.
• Remember to always include the constant of integration [latex]C[/latex].
• Verify your results by differentiating the antiderivative.
• Use the sum/difference and constant multiple rules to break down complex integrals.
• Be aware that the integral of a product is not the product of the integrals.
• Memorize the common indefinite integrals, especially trigonometric and exponential functions.
• When integrating rational functions, look for opportunities to use substitution or partial fractions.

Initial-Value Problems

• Practice solving basic differential equations before tackling initial-value problems.
• Use the initial condition to find the specific value of [latex]C[/latex] for the particular solution.
• Draw a diagram or sketch a graph to visualize the problem when possible.
• Always check your solution by substituting it back into both the differential equation and the initial condition.
• Pay attention to units, especially in applied problems.
• Practice interpreting the physical meaning of your mathematical solutions.
• Remember that some initial-value problems may have no solution or multiple solutions.