Calculus

1a Integral Calculus

e = 2.718281828459045235...

Consider the exponential function of y = aen. Uses sliders to change the values of A. and K. How does changing each value of affect the graph?

The value of a tells us where the graph crosses the y-axis, because a is the value of y when t = 0.

The parameter k is called the continuous growth rate. Provided a is positive, if k is positive, the graph increases when read from left to right. The greater the value of k, the more rapidly the graph increases. When k is negative, the graph decreases when read from left to right. The more negative the value of k, the more rapidly the graphs decreases.

base e

1.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This knowledge includes one-sided limits, infinite limits, and limits at infinity. Students know the definition of convergence and divergence of a function as the domain variable approaches either a number or infinity:

1.1 Students prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions.

1.2 Students use graphical calculators to verify and estimate limits.

1.3 Students prove and use special limits, such as the limits of (sin(x))/x and (1-cos(x))/x as x tends to 0.

2.0 Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.

3.0 Students demonstrate an understanding and the application of the intermediate value theorem and the extreme value theorem.

4.0 Students demonstrate an understanding of the formal definition of the derivative of a function at a point and the notion of differentiability:

4.1 Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function.

4.2 Students demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change. Students can use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function.

4.3 Students understand the relation between differentiability and continuity.

4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions.

5.0 Students know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.

6.0 Students find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of problems in physics, chemistry, economics, and so forth.

7.0 Students compute derivatives of higher orders.

8.0 Students know and can apply Rolle's theorem, the mean value theorem, and L'Hôpital's rule.

9.0 Students use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing.

10.0 Students know Newton's method for approximating the zeros of a function.

11.0 Students use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts.

12.0 Students use differentiation to solve related rate problems in a variety of pure and applied contexts.

13.0 Students know the definition of the definite integral by using Riemann sums. They use this definition to approximate integrals.

14.0 Students apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals.

15.0 Students demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives.

16.0 Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work.

17.0 Students compute, by hand, the integrals of a wide variety of functions by using techniques of integration, such as substitution, integration by parts, and trigonometric substitution. They can also combine these techniques when appropriate.

18.0 Students know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals.

19.0 Students compute, by hand, the integrals of rational functions by combining the techniques in standard 17.0 with the algebraic techniques of partial fractions and completing the square.

20.0 Students compute the integrals of trigonometric functions by using the techniques noted above.

21.0 Students understand the algorithms involved in Simpson's rule and Newton's method. They use calculators or computers or both to approximate integrals numerically.

22.0 Students understand improper integrals as limits of definite integrals.

23.0 Students demonstrate an understanding of the definitions of convergence and divergence of sequences and series of real numbers. By using such tests as the comparison test, ratio test, and alternate series test, they can determine whether a series converges.

24.0 Students understand and can compute the radius (interval) of the convergence of power series.

25.0 Students differentiate and integrate the terms of a power series in order to form new series from known ones.

26.0 Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term.

27.0 Students know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems.

1b Differential Calculus

NOTE: The calculus in this textbook is presented with the same level of depth and rigor as are entry-level college and university calculus courses, providing a complete college curriculum in one variable calculus. Treating nothing lightly, it makes the point of taking sufficient time to covering all the bases in an understandable, comprehensible and applicable manner, for example, spending substantial time on both differential equations and infinite sequences and series.

1a Functions

functions

NOTE: Make sure that you are familiar with set description and interval notation. Brackets mean the the enclosed values are included in the domain or range, whichever applies.

Parentheses, on the other hand, indicate that an endpoint is NOT included.

In other words, the domain is all of the values that x can be, and the range is all of the values tht y can be, given x.

LESSON 1

A logarithm with the vase can be rewritten as a logarithms with base e or with base 10. The formulas to the right show properties that allow you to rewrite logarithms.

You can use these formulas to evaluate iog823.

Using the first formula, log823 is equal to log 23 divided by log 8, which is approximately 1.508.

You also could have used his second formula to rewrite log823 as In23 divided by In8, which would have given you the same answer.

So then, calculus is the mathematics that helps us deal with calculating information about changing quantities. It was developed several hundred years ago, by Newton and Liebniz, to deal with such problems as finding the total distance traveled by an object with changing velocity, or finding the velocity of an object with changing position.

Suppose that a ball has been thrown into the air. Its height, in feet, after tea seconds is given by the equation: h = –16t² + 40t + 5.

To find the height at any time, t, we input the value of t into the equation.

When we do this, we obtained an output that is the height value. So time, t, is the input, and height, h, is the output.

Look at the graph of height versus time. Notice that there is exactly one corresponding height value for each time value.

Since each input value is assigned to exactly one output values, the graph confirms that height is a function of time, so, we say that the height of the ball is a function of time.

In general, any relationship that has only one output value for each input value is a function.

You saw that the height of a ball thrown in the air is a function of the time since it has been thrown. Can we also say that time is a function of high? In other words, is there exactly 1 time value assigned to each fight value?

Look at the graph. Noticed that Lee H. equals 21, he can either be 0.5 or 2. There are two possible outfits for this in place. So time is not a function of high.

In general, any relationship that has only one output value for each input value is a function. For a detailed explanation of functions click on this link.

A graph represents a function if any vertical line intersects the graph only once. This rule is called the vertical line test. (The input value is represented by the x coordinate and the output value, or function of x, is represented by the y coordinate.) The vertical line rule demonstrates that if a ball is thrown in the air, its height is a function of time, but time is not a function of height.

FUNCTION NOTATION

The statement h is a function of t can be shortened to h equals f of t, or shorter yet h = f(t), where h is the y coordinate, t is the x coordinate, and f is the "rule" or computation that is performed on x in order to generate y. In other words, whatever varible happens to be inside of the parentheses is actually representing the x coordinate.

Sometime you will be given the output value and required to find the corresponding input value. In such cases, the quadratic formula is one means of obtaining the solution (for example, the two points at which t satisfies the equation f(t) = 25.

Also, we call the solution of the equation f(t) = 0 a root or a zero of the function (just for your information).

The set of input values for which the function is definded (the "used" portion along the x-axis) is called the domain of the function, and the corresponding set of output values (along the "used" portion of the y-axis) is called the range of the function.

Use the graph to answer the following:

Is it true that f(0) = 5

Is it true that f(21) = 2

Is it true that f(5) = 0

Is it true that f(2) = 21

Does f (1.5) = 29 tell us that 1.5 seconds after the ball was thrown, its height was 29 meters -- or does it tell us that 29 seconds after the ball was thrown, its height was 1.5 meters?

Lets see how you can evaluate f (1.5) using the graph or the equation for f (t).

To use the graph to find an approximate value for f (1.5) first find the link on the graph with a t-coordinate of 1.5.

You can see that the vertical cordon of his point is slightly less than 30 feet. So the value of f (1.5) is approximately 28 or 29.

You can also evaluate f (1.5) by substituting 1.5 for t indict equations. This method gives us the exact value for f (1.5) which is 29 feet.

Another method for evaluating f (1.5) uses your calculator. Make sure that you can use your calculator to evaluate the value of a function for a given conflict. If you don't know how to do so, referred to that Making a Table section in the "Using Your Graphing Calculator" lesson.

You have seen how to evaluate the height of the ball at any time using the graph for the equation. Often the situation is reversed; we know the output value and we want to find the corresponding infant values.

Lets see how to find the time in the fall is 25 feet above the ground or he

In this case, we need to find in the values that he is an outfit of 25. In other words, we need to find t when f(t) = 25.

We can use the grass to estimate the solution. To do so, you will first find the link on the graph that have a vertical coordinate of 25.

You can see that there are two such flights. Has a t-coordinate of approximately 0.7, and the other has a t-coordinate of approximately 1.8.

To find the exact solutions, we can solve the equation algebraicly.

So, the height of the ball is 25 feet at two different times. Once we t = 0.691 seconds, and once when t = 1.809 seconds.

And of course, another method is using your calculator.

Recall tha we defined h to be the height of a ball t seconds after it has been thrown into the air. Although h is a function of t, the value of h is not defined for every possible value of t.

For example, it makes no sense to talk about the height of the ball when t = -2 or t = 4. So, although h is a function of t, this function is only defined for certain values of t.

Also, notice that h, that outfit value of the function, can only attain values between zero and 30.

Is set up in the values for which the function is the fine is called the domain of the function. The corresponding set the outfit values is called the range of the function.

So, the domain of the function in our example is the set of all real numbers between zero and 2.619. This set can also be described using interval notation, as shown.

Racket next 20 means that zero is included in the domain. Similarly, the bracket next to 2.6 and 19 means that 2.619 is included in the domain.

The range of the function is this set of all real numbers between zero and 30. Again, the brackets mean that zero and 30 are included in the range.

Now, consider the function h(x), whose graph is shown. We want to find the domain and range of this function.

By looking at the graph, you can see that the domain of this function is all real numbers. Is set of real numbers can be represented as the open interval from negative infinity to infinity.

The parentheses next to the infinity symbols mean that infinity and negative infinity are not included in the domain. This is because these values are not actually numbers.

Now let's find the range of h(x). The range of this function is all real numbers greater than zero.

You have seen how to solve equations such as 10 = 150 graphically. Now, you will learn another technique for solving such equations.

To solve the x, notice that we can rewrite this equation as a logarithm. So, x is equal to log 150.

To find the logarithm base 10 of a number, you can use the "log" button on your calculator.

log 150 is approximately 2.176 which means that 10 is approximately equal to 150.

Now consider the equation 100 ∙ 3 = 3500. and notice that the exponential firm in this equation has a base of three, not 10.

In order to solve this equation can use the properties of logarithms shown to the right to help us write his equation using logarithms.

First, we divide full-sized by 100.

Now, according to the first poverty, we can take the logarithms base 10 of both sides.

Using the fourth poverty in the list, we can rewrite the left side of equations as x ∙ log3.

Using a calculator to devalue with these logarithms, we find that x is approximately 3.236.

You can use this procedure to solve any exponential equations. First rewrite the equations of that is in the form bx - c. Then take the law or In of both sides, and use the properties of logarithms to solve for the variable.

time in seconds

height in feet

h = 16t² + 40t + 5

time in seconds

height in feet

h = 16t² + 40t + 5

LESSON 1: Definition of a function, Vertical line test,Function notation, Finding input and output 0values

Exponential Functions

Absolute value functions are all piecewise defined functions. (They are always shaped like a "v"?)

The irrational number e is oftern used as the base for exponential functions. Base e is so important that it is called the natural base. Exponential equations in calculus are much simpler to work with when they have base e.

Because e is a number between two and three, the graph of y = e' lies between the grass of y = 2' and y = 3'

2.176