Calculus

Worked examples – Limits

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Worked Examples – Limits The indeterminate forms of limits $ displaystyle infty cdot infty ,infty cdot 0,frac{infty }{infty },frac{0}{0},frac{infty }{0},infty +infty ,infty -infty {{,1}^{infty }}{{,0}^{infty }},{{infty }^{0}}$ The indeterminate forms of limits $displaystyle infty cdot infty ,infty cdot 0,frac{infty }{infty },frac{0}{0},frac{infty }{0},$ $displaystyle infty +infty ,infty -infty ,{{1}^{infty }},{{0}^{infty }},{{infty }^{0}}$ Important Limits $ displaystyle […]

Examples of calculating the derivative

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Examples of calculating the derivative Example 1: Find the derivative of the functions. Applying the constant rule: If $ displaystyle y=c$ then $ displaystyle y’=0$ The Power function rule: If $ displaystyle y=a{{x}^{n}}$ then $ displaystyle y’=an{{x}^{{n-1}}}$ a) $ displaystyle y=3{{x}^{7}}$ $ displaystyle y’=(3cdot 7){{x}^{{7-1}}}=21{{x}^{6}}$ b) $ displaystyle y=-4$ $ displaystyle y’=(-4)’=0$ c) $ displaystyle y=2{{x}^{{-1}}}$ $ […]

Derivative Rules

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Derivative Rules We know that if $displaystyle y=f(x)$ then the derivative is defined to be $displaystyle f'(x)=underset{{hto 0}}{mathop{{lim }}},frac{{f(x+h)-f(x)}}{h}$ Some notions we use when we write the derivative are: $ displaystyle y’=f'(x)=frac{{df}}{{dx}}=frac{{dy}}{{dx}}$ Determining the derivative of a function using the definition sometimes it  requires a lot of work and it’s easy to make mistakes so […]

The derivative using the limit definition

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The derivative using the limit definition Definition The derivative of the function $ displaystyle f$ at the point a is the limit when $ displaystyle hto 0$ of the function, if this limit exists. We label it f´(a) and $displaystyle f'(a)=underset{{hto 0}}{mathop{{lim }}},frac{{f(a+h)-f(a)}}{h}$ When the function$ displaystyle f$ is derivative on the point $ displaystyle […]

Combining Functions

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Combining Functions Combining Functions means performing basic arithmetic operations like addition, subtraction, multiplication and division with functions. Given two functions $ displaystyle f(x)$ and $ displaystyle g(x)$ we define: 1. The sum of two functions $displaystyle (f+g)(x)=f(x)+g(x)$ 2. The difference of two functions $ displaystyle (f-g)(x)=f(x)-g(x)$ 3. The product of two functions $ displaystyle (ftimes […]

Theorems on Limits

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Theorems on Limits It can be rather tedious to apply the $displaystyle varepsilon $ and $displaystyle delta $ limit test to individual functions. By remembering some basic theorems about limits we can avoid the some of this repetitive work. We shouldn’t forget that if a limit exists it is always unique.  “The Uniqueness of a Limit” […]

Introduction to Limits

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Introduction to Limits Numerical and Graphical approach to limits Numerical Approach Let’s take a function f(x) and see how the values of the functions change when x takes values closer to a specific number. Example: Let f(x)=3x+1 and calculate f(x) as x takes values closer to 1, but not exactly the value at 1. We first […]

The Limit of a Function

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The limit of a function Let f be a function and let c be a real number. We do not require that f be defined ar c but we do require that f be defined at least on a set of the form (c-p,c) U (c,c+p) with p>0). To say that $displaystyle underset{{xto c}}{mathop{{lim }}},f(x)=l$ is […]

The graph of a function

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The graph of a function We know that the graph of a function is the set of all points of the plan xOy that have like abscissa the faces (elements of the domain sets) and like ordinate they have the corresponding value of the function. The graph of the numeric function is the illustration of the […]

Function

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Function The relation f with first set A and second set B is called a function when every element of A is paired with only one element of B.  f:A→B We can write it based on the way we label the sets. You may find it written as f:X→Y If in the functional relation f with first set […]

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