Calculus 1

SEQUENCES

• Index of sequences - The index of the sequence tells us which member of the sequence we are looking at.
• Limits of sequences - One of the most important properties of a sequence is what happens to it when we look at its members that are further and further away.
• Common sequences - Exponential, polynomial and radical sequences.
• Limits and operations - Well, the point of these tables is to help us navigate among the various types of limits.
• Find the limits - Divide both the numerator and the denominator by the strongest member of the denominator.
• Convergent sequences - A sequence is called convergent if there is a real number that is the limit of the sequence.
• Divergent sequences - A sequence is divergent if it tends to infinity, but it is also divergent if it doesn’t tend to anywhere at all.
• Oscillating sequences - Sequences that tend to nowhere are always oscillating sequences.
• Squeeze Theorem - We need a sequence that is less than the original sequence, and another one that is greater than the original sequence.
• Estimation - When finding the lower estimate, we omit everything but the strongest term, and when finding the upper estimate, we replace all terms with the strongest term.

LIMIT OF THE FUNCTION

• Limits - At the same time, as x approaches a, the values of the function approach B.
• Computing limits - The first step is to substitute into the function. Let's see what we get. If the result is not interpretable, then we are in trouble. In this case there are usually three possibilities.
• Limits of rational functions - We have to factorize both the numerator and the denominator.
• Limits of trigonometric functions - Let's talk about the limits of trigonometric functions. Here we have a few exciting cases.

CONTINUITY

• Continuity of functions - We call the function continuous at x where the limit and the value of the function are the same.
• Discontinuity - If there is a finite limit at a, but it is not the same as the value of the function, or the limit don’t even exists then the function is discontinuity at a.
• Removable discontinuity - If there is a finite limit at a, but it is not the same as the value of the function.
• Jump discontinuity - If the left and right-side limits at a are two different numbers, then the discontinuity is not removable.
• Infinite discontinuity - If the left and right-side limits are not even finite at a, then the function has an infinite discontinuity.
• Oscillating discontinuity - There are some rather pathological cases, where the right or left-side limits don’t even exist.

DIFFERENTATION

• The definition of the derivative - The slope of the tangent is the derivative. Let's see the story.
• Differentation formulas - Here we will start introducing some of the differentiation formulas.
• Chain rule - The differentiation rule for composite functions
• The precise definition of the derivative -
• Differentation -

APPLICATIONS OF DERIVATIVES

• The first derivative and the monotonicity - If the derivative of the function is positive, then the function increases, if the derivative is negative, then the function decreases.
• The second derivative and the concavity - The "mood of the function" is indicated by its second derivative. If the second derivative is negative, then the function is concave, if positive, then it is convex.
• Stationary point and the sign of derivative - We check when the derivative is positive and when negative.
• The shape of graph - We will look at the information about the graph of a function that the first and second derivatives can tell us.
• Business Applications - The demand function, the revenues function and the quantity function.
• L'Hospital's Rule -
• Taylor Polinomyals and Taylor series -

INTEGRATION

• Definite and indefinite integrals - Definite integration deals with calculating the area under the curve of a function. The indefinite integration works a totally different way. The indefinite integral of f(x) is a function that is called the primitive function.
• Primitive function - The symbol for the primitive function is F(x), and its main property is that if we differentiate it, we get f(x).
• Fundamental theorem of calculus - The theorem that describes this relationship is one of the most important theorems in the entire history of mathematics. It was developed at the end of the 1600s by two men at the same time: an English physicist named Newton and a German philosopher named Leibniz.
• Basic integration formulas - We start looking for primitive functions by remembering the derivatives of a few important functions.
• Integration rules - We will have to know which rule to use, and it is not recommended to decide by tossing a coin.
• Integration by parts - The purpose of partial integration is to turn a complicated integral into a simpler one.
• Integration of composite functions - This theorem is really about integrating composite functions. But unfortunately composite functions are a bit problematic, as computing their integral is usually a quite hopeless undertaking. Let's see.
• Integration by substitution - The main idea of integration by substitution is that we replace an expression by u, hoping that this way we may be able to solve the problem.
• Partial fractions - Integrating rational functions is an extremely entertaining activity. We will start by developing our ability to integrate so called partial fractions.
• Integrating rational functions - We can easily integrate any rational function. All we have to do is decompose it into partial fractions, and then integrate the partial fractions.
• Polynomial division - This polynomial division is just like long division we learned in elementary school, but that much funnier.
• Integration of trigonometric functions - Integration of trigonometric terms is not easy. We will check out a few simpler tricks and the most important methods.
• Tangent half-angle substitution - One of the strangest cases of integration by substitution is when u= tan(x/2). We use this if sinx and cosx is included in first-degree form in the fraction.
• Definite integral - Definite integration deals with calculating the area under the curve of a function.
• Improper integrals - These integrals that stretch into infinity are called improper integrals.