# Calculus Explained

Note that some sections can have additional issues than others and a few can have additional or less of a range of issues. **Most sections ought to have a spread of problem levels within the issues though this may vary from section to section.**

Here could be a listing of sections that follow issues are written similarly as a quick description of the fabric coated within the notes for that specific section.

Review — during this chapter we have a tendency to provides a temporary review of designated topics from pure mathematics and Trig that ar very important to extant a Calculus course. enclosed ar Functions, Trig Functions, determination Trig Equations and Equations, Exponential/Logarithm Functions and determination Exponential/Logarithm Equations.

Functions — during this section we’ll cowl operate notation/evaluation, deciding the domain and vary of a operate and performance composition.

Inverse Functions — during this section we’ll outline Associate in Nursing mathematical function and therefore the notation used for inverse functions. we’ll conjointly discuss the method for locating Associate in Nursing mathematical function.

Trig Functions — during this section we’ll provides a fast review of trig functions. we’ll cowl the essential notation, relationship between the trig functions, the proper triangle definition of the trig functions. we’ll conjointly cowl analysis of trig functions similarly because the unit circle (one of the foremost necessary concepts from a trig class!) and the way it are often wont to valuate trig functions.**Solving Trig Equations** — during this section we’ll discuss the way to solve trig equations. The answers to the equations during this section can all be one in every of the “standard” angles that almost all students have memorized when a trig category. However, the method used here are often used for any answer in spite of it being one in every of the quality angles or not.

Solving Trig Equations with Calculators, half I — during this section {we can|we’ll|we are going to} discuss determination trig equations once the solution will (generally) need the employment of a calculator (i.e. they aren’t one in every of the quality angles). Note but, the method used here is just like that for once the solution is one in every of the quality angles. the sole distinction is that the answers in here are often a touch untidy because of the necessity of a calculator. enclosed could be a temporary discussion of inverse trig functions.

Solving Trig Equations with Calculators, half II — during this section we’ll continue our discussion of determination trig equations once a calculator is required to induce the solution. The equations during this section tend to be a touch trickier than the “normal” trig equation and don’t seem to be continually coated in an exceedingly trig category.

Exponential Functions –In this section we’ll discuss exponential functions. we’ll cowl the essential definition of Associate in Nursing function, the natural function, i.e.

e

x

, similarly because the properties and graphs of exponential functions

Logarithm Functions — during this section we’ll discuss exponent functions, analysis of logarithms and their properties. we’ll discuss several of the essential manipulations of logarithms that ordinarily occur in Calculus (and higher) categories. enclosed could be a discussion of the natural (

ln

(

x

)

) and customary exponent (

log

(

x

)

) similarly because the amendment of base formula.

Exponential and exponent Equations — during this section we’ll discuss numerous strategies for determination equations that involve exponential functions or exponent functions.

Common Graphs — during this section we’ll do a really fast review of the many of the foremost common functions and their graphs that usually show up in an exceedingly Calculus category.

**Limits **— during this chapter we have a tendency to introduce the idea of limits. we’ll discuss the interpretation/meaning of a limit, the way to valuate limits, the definition and analysis of one-sided limits, analysis of infinite limits, analysis of limits at eternity, continuity and therefore the Intermediate worth Theorem. we’ll conjointly provides a temporary introduction to an exact definition of the limit and the way to use it to guage limits

Tangent Lines and Rates of amendment — during this section we’ll introduce 2 issues that we’ll see time and once more during this course : Rate of amendment of a operate and Tangent Lines to functions. each of those issues are wont to introduce the idea of limits, though we can’t formally offer the definition or notation till subsequent section.

The Limit — during this section we’ll introduce the notation of the limit. we’ll conjointly take a abstract investigate limits and check out to induce a grasp on simply what they’re and what they will tell US. we’ll be estimating the worth of limits during this section to assist US perceive what they tell US. we’ll really begin computing limits in an exceedingly number of sections.

One-Sided Limits — during this section we’ll introduce the idea of one-sided limits. we’ll discuss the variations between one-sided limits and limits similarly as however they’re associated with one another.

Limit Properties — during this section we’ll discuss the properties of limits that we’ll ought to use in computing limits (as opposition estimating them as we’ve done to the present point). we’ll conjointly cipher one or two of basic limits during this section.

Computing Limits — during this section we’ll appearance at many varieties of limits that need some work before we will use the limit properties to cipher them. we’ll conjointly investigate computing limits of piecewise functions and use of the Squeeze Theorem to cipher some limits.

Infinite Limits — during this section we’ll investigate limits that have a worth of eternity or negative eternity. We’ll conjointly take a quick investigate vertical asymptotes.

Limits At eternity, half I — during this section we’ll begin gazing limits at eternity, i.e. limits within which the variable gets terribly giant in either the positive or negative sense. we’ll consider polynomials and rational expressions during this section. We’ll conjointly take a quick investigate horizontal asymptotes.

Limits At eternity, half II — during this section we’ll continue covering limits at eternity. We’ll be gazing exponentials, logarithms and inverse tangents during this section.

Continuity — during this section we’ll introduce the idea of continuity and the way it relates to limits. we’ll conjointly see the Intermediate worth Theorem during this section and the way it are often wont to confirm if functions have solutions in an exceedingly given interval.

The Definition of the Limit — during this section we’ll provides a precise definition of many of the bounds coated during this section. we’ll work many basic examples illustrating the way to use this precise definition to cipher a limit. We’ll conjointly provides a precise definition of continuity..

*Derivatives — during this chapter we have a tendency to introduce Derivatives. we have a tendency to cowl the quality derivatives formulas as well as the merchandise rule, quotient rule and chain rule similarly as derivatives of polynomials, roots, trig functions, inverse trig functions, hyperbolic functions, exponential functions and exponent functions. we have a tendency to conjointly cowl implicit differentiation, connected rates, higher order derivatives and index differentiation.*

The Definition of the spinoff — during this section we have a tendency to outline the spinoff, offer numerous notations for the spinoff and work some issues illustrating the way to use the definition of the spinoff to truly cipher the spinoff of a operate.

Interpretation of the spinoff — during this section we have a tendency to offer many of the additional necessary interpretations of the spinoff. we have a tendency to discuss the speed of amendment of a operate, the speed of a moving object and therefore the slope of the tangent line to a graph of a operate.

Differentiation Formulas — during this section we have a tendency to offer most of the overall spinoff formulas and properties used once taking the spinoff of a operate. Examples during this section concentrate totally on polynomials, roots and additional usually variables raised to powers.

Product and Quotient Rule — during this section we’ll offer 2 of the additional necessary formulas for differentiating functions. we’ll discuss the merchandise Rule and therefore the Quotient Rule permitting US to differentiate functions that, up to the present purpose, we have a tendency to were unable to differentiate.

Derivatives of Trig Functions — during this section we’ll discuss differentiating trig functions. Derivatives of all six trig functions ar given and that we show the derivation of the spinoff of

sin

(

x

)

and

tan

(

x

)

.

Derivatives of Exponential and exponent Functions — during this section we have a tendency to derive the formulas for the derivatives of the exponential and exponent functions.

Derivatives of Inverse Trig Functions — during this section we have a tendency to offer the derivatives of all six inverse trig functions. we have a tendency to show the derivation of the formulas for inverse trigonometric function, inverse cos and inverse tangent.

Derivatives of Hyperbolic Functions — during this section we have a tendency to outline the hyperbolic functions, offer the relationships between them and a few of the essential facts involving hyperbolic functions. we have a tendency to conjointly offer the derivatives of every of the six hyperbolic functions and show the derivation of the formula for hyperbolic trigonometric function.

Chain Rule — during this section we have a tendency to discuss one in every of the additional helpful and necessary differentiation formulas, The Chain Rule. With the chain rule out hand we’ll be ready to differentiate a far wider type of functions. As {you can|you’ll|you may} see throughout the remainder of your Calculus courses a good several of derivatives you’re taking will involve the chain rule!

Implicit Differentiation — during this section we’ll discuss implicit differentiation. Not each operate are often expressly written in terms of the variable, e.g. y = f(x) and however we’ll still ought to understand what f’(x) is. Implicit differentiation can enable US to seek out the spinoff in these cases. Knowing implicit differentiation can enable US to try to to one in every of the additional necessary applications of derivatives, connected Rates (the next section).*Related Rates — during this section we’ll discuss the sole application of derivatives during this section, connected Rates. In connected rates issues we have a tendency to ar offer the speed of amendment of 1 amount in an exceedingly drawback and asked to work out the speed of 1 (or more) quantities within the drawback. this is {often|this can be} often one in every of the harder sections for college kids. we have a tendency to work quite an few issues during this section thus hopefully by the top of this section you may get an honest understanding on however these issues work.*

Higher Order Derivatives — during this section we have a tendency to outline the idea of upper order spinoffs and provides a fast application of the second order derivative and show however implicit differentiation works for higher order derivatives.You can refer to https://integralcalculator.info/ for easy calculations.

Logarithmic Differentiation — during this section we’ll discuss index differentiation. index differentiation offers another methodology for differentiating merchandise and quotients (sometimes easier than mistreatment product and quotient rule). additional significantly, however, is that the indisputable fact that exponent differentiation permits US to differentiate operates that ar within the sort of one operate raised to a different function, i.e. there ar variables in each the bottom and exponent of the operate.