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Integration is the inverse of differentiation.


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Even though derivatives are fairly straight forward, integrals are Differentiation is a method to calculate the rate of change or the slope at a point on the graph ; we will not Sign In Sign in with Office Sign in with Facebook. Join 90 million happy users! Sign Up free of charge:. Join with Office Join with Facebook. I agree to the terms and conditions Create my account. Transaction Failed!

Please try again using a different payment method. Subscribe to get much more:. Both of these problems will be used to introduce the concept of limits, although we won't formally give the definition or notation until the next section. The Limit — In this section we will introduce the notation of the limit. We will also take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us. We will be estimating the value of limits in this section to help us understand what they tell us. We will actually start computing limits in a couple of sections.

One-Sided Limits — In this section we will introduce the concept of one-sided limits.

A Gentle Introduction To Learning Calculus – BetterExplained

We will discuss the differences between one-sided limits and limits as well as how they are related to each other. We will also compute a couple of basic limits in this section.

Computing Limits — In this section we will looks at several types of limits that require some work before we can use the limit properties to compute them. We will also look at computing limits of piecewise functions and use of the Squeeze Theorem to compute some limits. Infinite Limits — In this section we will look at limits that have a value of infinity or negative infinity.

We will concentrate on polynomials and rational expressions in this section. Continuity — In this section we will introduce the concept of continuity and how it relates to limits. We will also see the Intermediate Value Theorem in this section and how it can be used to determine if functions have solutions in a given interval.

The Definition of the Limit — In this section we will give a precise definition of several of the limits covered in this section. We will work several basic examples illustrating how to use this precise definition to compute a limit. The Definition of the Derivative — In this section we define the derivative, give various notations for the derivative and work a few problems illustrating how to use the definition of the derivative to actually compute the derivative of a function.

Calculating velocities and slopes

Interpretation of the Derivative — In this section we give several of the more important interpretations of the derivative. We discuss the rate of change of a function, the velocity of a moving object and the slope of the tangent line to a graph of a function. Differentiation Formulas — In this section we give most of the general derivative formulas and properties used when taking the derivative of a function. Examples in this section concentrate mostly on polynomials, roots and more generally variables raised to powers.

Product and Quotient Rule — In this section we will give two of the more important formulas for differentiating functions. We will discuss the Product Rule and the Quotient Rule allowing us to differentiate functions that, up to this point, we were unable to differentiate. Derivatives of Trig Functions — In this section we will discuss differentiating trig functions.

Trigonometric Integrals

Derivatives of Exponential and Logarithm Functions — In this section we derive the formulas for the derivatives of the exponential and logarithm functions. Derivatives of Inverse Trig Functions — In this section we give the derivatives of all six inverse trig functions. We show the derivation of the formulas for inverse sine, inverse cosine and inverse tangent.

Derivatives of Hyperbolic Functions — In this section we define the hyperbolic functions, give the relationships between them and some of the basic facts involving hyperbolic functions. We also give the derivatives of each of the six hyperbolic functions and show the derivation of the formula for hyperbolic sine. Chain Rule — In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule.

With the chain rule in hand we will be able to differentiate a much wider variety of functions. As you will see throughout the rest of your Calculus courses a great many of derivatives you take will involve the chain rule! Implicit Differentiation — In this section we will discuss implicit differentiation.

Not every function can be explicitly written in terms of the independent variable, e. Implicit differentiation will allow us to find the derivative in these cases. Knowing implicit differentiation will allow us to do one of the more important applications of derivatives, Related Rates the next section. Related Rates — In this section we will discuss the only application of derivatives in this section, Related Rates. In related rates problems we are give the rate of change of one quantity in a problem and asked to determine the rate of one or more quantities in the problem.

This is often one of the more difficult sections for students. We work quite a few problems in this section so hopefully by the end of this section you will get a decent understanding on how these problems work. Higher Order Derivatives — In this section we define the concept of higher order derivatives and give a quick application of the second order derivative and show how implicit differentiation works for higher order derivatives. Logarithmic Differentiation — In this section we will discuss logarithmic differentiation.

Logarithmic differentiation gives an alternative method for differentiating products and quotients sometimes easier than using product and quotient rule. More importantly, however, is the fact that logarithm differentiation allows us to differentiate functions that are in the form of one function raised to another function, i. Critical Points — In this section we give the definition of critical points. Critical points will show up in most of the sections in this chapter, so it will be important to understand them and how to find them.

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We will work a number of examples illustrating how to find them for a wide variety of functions. Minimum and Maximum Values — In this section we define absolute or global minimum and maximum values of a function and relative or local minimum and maximum values of a function.

We also give the Extreme Value Theorem and Fermat's Theorem, both of which are very important in the many of the applications we'll see in this chapter. Finding Absolute Extrema — In this section we discuss how to find the absolute or global minimum and maximum values of a function. In other words, we will be finding the largest and smallest values that a function will have.

The Shape of a Graph, Part I — In this section we will discuss what the first derivative of a function can tell us about the graph of a function. The first derivative will allow us to identify the relative or local minimum and maximum values of a function and where a function will be increasing and decreasing. We will also give the First Derivative test which will allow us to classify critical points as relative minimums, relative maximums or neither a minimum or a maximum.

Calculus Applied!

The Shape of a Graph, Part II — In this section we will discuss what the second derivative of a function can tell us about the graph of a function. The second derivative will allow us to determine where the graph of a function is concave up and concave down. The second derivative will also allow us to identify any inflection points i. We will also give the Second Derivative Test that will give an alternative method for identifying some critical points but not all as relative minimums or relative maximums.

With the Mean Value Theorem we will prove a couple of very nice facts, one of which will be very useful in the next chapter. We will discuss several methods for determining the absolute minimum or maximum of the function. Examples in this section tend to center around geometric objects such as squares, boxes, cylinders, etc. More Optimization Problems — In this section we will continue working optimization problems.

The examples in this section tend to be a little more involved and will often involve situations that will be more easily described with a sketch as opposed to the 'simple' geometric objects we looked at in the previous section. Linear Approximations — In this section we discuss using the derivative to compute a linear approximation to a function.

We can use the linear approximation to a function to approximate values of the function at certain points. While it might not seem like a useful thing to do with when we have the function there really are reasons that one might want to do this. We give two ways this can be useful in the examples. Differentials — In this section we will compute the differential for a function. We will give an application of differentials in this section.