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Essential Concepts

Limits at Infinity and Asymptotes

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

Newton’s Method

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

Antiderivatives

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

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

Key Equations

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

Glossary

antiderivative

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

end behavior

the behavior of a function as $x\to \infty$ and $x\to −\infty$

horizontal asymptote

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

indeterminate forms

when evaluating a limit, the forms $0/0$, $\infty / \infty$, $0 \cdot \infty$, $\infty -\infty$, $0^0$, $\infty^0$, and $1^{\infty}$ 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 $f(x)$ is the indefinite integral of $f$; we use the notation $\displaystyle\int f(x) dx$ to denote the indefinite integral of $f$

infinite limit at infinity

a function that becomes arbitrarily large as $x$ becomes large

initial value problem

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

iterative process

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

L’Hôpital’s rule

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

limit at infinity

the limiting value, if it exists, of a function as $x\to \infty$ or $x\to −\infty$

Newton’s method

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

oblique asymptote

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

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 $x$ 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 $\frac{0}{0}$ or $\frac{\infty}{\infty}$ 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 $0 \cdot \infty$ forms, try rewriting as $\frac{\text{term approaching 0}}{\frac{1}{\text{term approaching }\infty}}$.
• For $\infty - \infty$ 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 $\frac{\ln(\text{base})}{\frac{1}{\text{exponent}}}$ and $\frac{\text{exponent}}{\frac{1}{\ln(\text{base})}}$ 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 $x \to \infty$.
• 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 $f'(x)$ 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 $C$ when expressing the general antiderivative.
• Pay attention to the domain of the function when finding antiderivatives, especially for functions like $1/x$.
• 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 $C$.
• 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 $C$ 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.