Differential equations Differential equations show you relationships between rates of functions. Separable differential equations. Separable differential equations Separable differential equations are those in which the dependent and independent variables can be separated on opposite sides of the equation. Parametric equations We discuss the basics of parametric curves.
Calculus and parametric curves We discuss derivatives of parametrically defined curves. Introduction to polar coordinates. Introduction to polar coordinates Polar coordinates are coordinates based on an angle and a radius. Gallery of polar curves We see a collection of polar curves. Derivatives of polar functions. Derivatives of polar functions We differentiate polar functions. Integrals of polar functions.
Integrals of polar functions We integrate polar functions. Working in two and three dimensions. Working in two and three dimensions We talk about basic geometry in higher dimensions. Vectors Vectors are lists of numbers that denote direction and magnitude.
The Dot Product The dot product is an important operation between vectors that captures geometric information. Projections and orthogonal decomposition Projections tell us how much of one vector lies in the direction of another and are important in physical applications.
The cross product The cross product is a special way to multiply two vectors in three-dimensional space. Jason Miller and Jim Talamo. Recall that in order for a function defined on to be bounded on , there must be some real number such that This says that the outputs of are trapped between the horizontal lines and ; the output values cannot become arbitrarily large in either the positive or negative sense on. An integral is called an improper integral if one of, or both, of the conditions hold: The interval of integration is infinite.
The function is unbounded on the interval of integration. Which of the following integrals are improper according to the previous definition? Note that is continuous everywhere, so it is bounded on any finite interval. Hence is proper.
Evaluate the improper integral We interpret definite integrals as giving us the net area underneath the graph of the function over the given interval. Compute The previous example shows us we should consider this expression to be the limit of a definite integral in the following manner: We calculate the definite integral This gives the area under on the interval. Let be a continuous function on. Let be any real number.
The improper integral converges if and only if both and independently converge. Determine whether the integral converges or diverges. If it converges, determine the value of the integral. Here both of the bounds of integration involve so to reinterpret the integral in terms of limits of definite integrals, we must first break up the integral into two pieces.
Now we must look at each integral separately. Compute: Write with me,. Compute: Write with me: Let us evaluate these limits one at a time: We can stop here. In the last example, what would have happened if we had tried to compute the indefinite integral with a single limit? Since sine is an odd function on any bounded interval , we would have found this integral to be.
However, We must compute two limits to evaluate this integral correctly. Compute: It may not be obvious at first glance that this integral is improper, but it is. Here the problem is that the function is unbounded on because gets arbitrarily large as approaches from the right That is, there is a vertical asymptote at because.
From now on, we will need to be cautious when evaluating integrals to check whether the integrand is bounded on the region of integration. Suppose lies strictly between and. Calculus with Power Series Taylor Series Taylor's Theorem Additional exercises 12 Three Dimensions 1. The Coordinate System 2. Vectors 3. The Dot Product 4. The Cross Product 5. Lines and Planes 6. Other Coordinate Systems 13 Vector Functions 1.
Space Curves 2. Calculus with vector functions 3. Arc length and curvature 4. Motion along a curve 14 Partial Differentiation 1. Functions of Several Variables 2. Limits and Continuity 3.
Partial Differentiation 4. The Chain Rule 5. Directional Derivatives 6. Higher order derivatives 7. Maxima and minima 8. Lagrange Multipliers 15 Multiple Integration 1. Volume and Average Height 2. Double Integrals in Cylindrical Coordinates 3. Moment and Center of Mass 4.
Surface Area 5. Triple Integrals 6. Cylindrical and Spherical Coordinates 7. Change of Variables 16 Vector Calculus 1. Vector Fields 2.
Line Integrals 3. The Fundamental Theorem of Line Integrals 4. Green's Theorem 5. Divergence and Curl 6. Vector Functions for Surfaces 7. For example, consider the function As seen in Figure and Figure , as the values become arbitrarily large. Therefore, On the other hand, as the values of are negative but become arbitrarily large in magnitude.
Informal We say a function has an infinite limit at infinity and write. We say a function has a negative infinite limit at infinity and write. Similarly, we can define infinite limits as. Earlier, we used the terms arbitrarily close , arbitrarily large , and sufficiently large to define limits at infinity informally. Although these terms provide accurate descriptions of limits at infinity, they are not precise mathematically. Here are more formal definitions of limits at infinity.
We then look at how to use these definitions to prove results involving limits at infinity. Formal We say a function has a limit at infinity, if there exists a real number such that for all there exists such that.
We say a function has a limit at negative infinity if there exists a real number such that for all there exists such that. Earlier in this section, we used graphical evidence in Figure and numerical evidence in Figure to conclude that Here we use the formal definition of limit at infinity to prove this result rigorously. Use the formal definition of limit at infinity to prove that. Let Let Therefore, for all we have.
We now turn our attention to a more precise definition for an infinite limit at infinity. Formal We say a function has an infinite limit at infinity and write. Similarly we can define limits as. Earlier, we used graphical evidence Figure and numerical evidence Figure to conclude that Here we use the formal definition of infinite limit at infinity to prove that result. Use the formal definition of infinite limit at infinity to prove that. Let Let Then, for all we have. Consider the power function where is a positive integer.
From Figure and Figure , we see that. Using these facts, it is not difficult to evaluate and where is any constant and is a positive integer. If the graph of is a vertical stretch or compression of and therefore. If the graph of is a vertical stretch or compression combined with a reflection about the -axis, and therefore. If in which case. For each function evaluate and. Let Find. We now look at how the limits at infinity for power functions can be used to determine for any polynomial function Consider a polynomial function.
As all the terms inside the parentheses approach zero except the first term. We conclude that. For example, the function behaves like as as shown in Figure and Figure. The end behavior for rational functions and functions involving radicals is a little more complicated than for polynomials.
In Figure , we show that the limits at infinity of a rational function depend on the relationship between the degree of the numerator and the degree of the denominator. To evaluate the limits at infinity for a rational function, we divide the numerator and denominator by the highest power of appearing in the denominator. This determines which term in the overall expression dominates the behavior of the function at large values of. For each of the following functions, determine the limits as and Then, use this information to describe the end behavior of the function.
Since we know that is a horizontal asymptote for this function as shown in the following graph. Therefore has a horizontal asymptote of as shown in the following graph. As the denominator approaches 1. As the numerator approaches As the numerator approaches Therefore whereas as shown in the following figure. Evaluate and use these limits to determine the end behavior of. Divide the numerator and denominator by. Before proceeding, consider the graph of shown in Figure. As and the graph of appears almost linear.
Although is certainly not a linear function, we now investigate why the graph of seems to be approaching a linear function. First, using long division of polynomials, we can write. Since as we conclude that.
Therefore, the graph of approaches the line as This line is known as an oblique asymptote for Figure. We can summarize the results of Figure to make the following conclusion regarding end behavior for rational functions.
Consider a rational function. In this case, we call an oblique asymptote. Determining End Behavior for a Function Involving a Radical Find the limits as and for and describe the end behavior of. Therefore, we divide the numerator and denominator by Then, using the fact that for for and for all we calculate the limits as follows:. Therefore, approaches the horizontal asymptote as and the horizontal asymptote as as shown in the following graph.
The six basic trigonometric functions are periodic and do not approach a finite limit as For example, oscillates between Figure. The tangent function has an infinite number of vertical asymptotes as therefore, it does not approach a finite limit nor does it approach as as shown in Figure.
Recall that for any base the function is an exponential function with domain and range If is increasing over If is decreasing over For the natural exponential function Therefore, is increasing on and the range is The exponential function approaches as and approaches 0 as as shown in Figure and Figure.
Recall that the natural logarithm function is the inverse of the natural exponential function Therefore, the domain of is and the range is The graph of is the reflection of the graph of about the line Therefore, as and as as shown in Figure and Figure.
Find the limits as and for and describe the end behavior of. To find the limit as divide the numerator and denominator by. As shown in Figure , as Therefore,. We conclude that and the graph of approaches the horizontal asymptote as To find the limit as use the fact that as to conclude that and therefore the graph of approaches the horizontal asymptote as. Find the limits as and for.
We now have enough analytical tools to draw graphs of a wide variety of algebraic and transcendental functions. Given a function use the following steps to sketch a graph of. Determine whether has any vertical asymptotes. Calculate Find all critical points and determine the intervals where is increasing and where is decreasing. Determine whether has any local extrema. Calculate Determine the intervals where is concave up and where is concave down.
Use this information to determine whether has any inflection points. The second derivative can also be used as an alternate means to determine or verify that has a local extremum at a critical point. We start by graphing a polynomial function. Sketch a graph of. It is not just that there is a vertical asymptote in the interval you are integrating over, it is also that that specific singularity "blows up too fast". It is not necessary that the integral will diverge if there are vertical asymptotes in the domain of integration.
When you observe this, you know something fishy is going on. The crux of the answer is that it is not necessary for an integral to diverge merely because of the presence a vertical asymptote in the domain of integration. Both the rate of growth of the integral about the asymptote and the sign of the values of it takes in that neighbourhood contribute to the convergence of the integral.
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