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## Free Math Homework Help Calculus

Calculus lies at the center of math and science. In simple terms, calculus involves the application of mathematics to study and understand change. Calculus traces its origins to the 17th century when notable scientists such as Sir Isaac Newton developed the discipline as a means of solving complex problems that could not be solved with more basic algebra concepts.

Calculus is easier for students to find interesting compared to other areas of math. Before we have some fun, let's review the basic topics that are normally considered part of college calculus that we can help with:

• FUNCTIONS AND MODELS
• LIMITS
• DERIVATIVES
• APPLICATIONS OF DIFFERENTIATION
• INTEGRALS
• APPLICATIONS OF INTEGRATION
• INVERSE FUNCTIONS
• TECHNIQUES OF INTEGRATION
• DIFFERENTIAL EQUATIONS
• PARAMETRIC EQUATIONS AND POLAR COORDINATES
• INFINITE SEQUENCES AND SERIES
• VECTORS AND THE GEOMETRY OF SPACE
• VECTOR FUNCTIONS
• PARTIAL DERIVATIVES
• MULTIPLE INTEGRALS
• VECTOR CALCULUS
• SECOND-ORDER DIFFERENTIAL EQUATIONS

There are a number of informative resources available to help your broaden your calculus knowledge base. A great place to get instructive articles in calculus is matharticles.com. If you haven't seen the MIT calculus videos, you've been missing out. These are essential for anyone interested in calculus.

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Calculus is the study of change, it basically analyses things that change and is a significant part  of Mathematics. Calculus is a branch of mathematics focused on limits, integrals, derivatives, functions and infinite series. Integral calculus and Differential calculus are the two main branches of this topic. Differential calculus is concerned with the study of rates at which a quantities change whereas Integral calculus gives information about the accumulation of quantities. These branches are connected with each other in respect of fundamental theorem. It is stated that calculus was founded in the 17th century and since then its concepts have been applied in many sectors including engineering, science, economics, computer science, medicine and others. Calculus is referred as the part of modern Mathematics.

Calculus is a widespread topic and it has three parts like ancient, medieval and modern calculus. Limits, derivative applications, solid of revolution are some important sub-topics that students are requested to learn intensely to get a thorough understanding of the topic. Some students find calculus tough, in such cases they are suggested to take online calculus help. TutorVista provides productive and informative sessions for calculus. To avail these online sessions designed for calculus, students need to follow some easy steps. They can choose their sub-topics and can take sessions at any preferred time. Moreover, they can take assistance in solving assessments and homework from our efficient online tutors.

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## Topics Covered in Calculus

• Functions Limits and Continuity
• Differentiation
• Differential Equations
• Indefinite Integrals
• Definite Integrals
• Applications of Derivatives
• Exponential and Logarithmic Series

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Below are some examples based on calculus:

### Solved Examples

Question 1: Find the derivative of f(x) = sin x using first principles.
Solution:

f(x) = sin x ... (1)

f(x + h) = sin (x + h) .....(2)

Subtract equation (1) from equation (2), we get

f(x + h) - f(x) = sin(x + h) - sin x

= 2 cos ($\frac{2x+h}{2}$) sin ($\frac{h}{2}$)    [ using identity sin A - sin B = 2 cos($\frac{A+B}{2}$) sin ($\frac{A-B}{2}$)

= 2 cos (x + $\frac{h}{2}$) sin ($\frac{h}{2}$)

$\frac{f(x + h) - f(x)}{h}$ = $\frac{1}{h}$ (2 cos (x + $\frac{h}{2}$) sin ($\frac{h}{2}$))

= $\frac{2 cos (x + \frac{h}{2}) sin (\frac{h}{2})}{2 \times \frac{h}{2}}$  [ Because $\frac{sin \frac{h}{2}}{\frac{h}{2}}$ ]

= cos(x + $\frac{h}{2}$)
$\lim_{h\rightarrow0}$ $\frac{f(x + h) - f(x)}{h}$ = $\lim_{h\rightarrow0}$  cos(x + $\frac{h}{2}$)

= cos x

=> Derivative of sin x is cos x.

Question 2: Solve $\int$ $\frac{log\ x}{x}$ dx

Solution:

Given: $\int$ $\frac{log\ x}{x}$ dx
Substitute log x = t, then $\frac{1}{x}$ dx = dt

Now $\int$ $\frac{log\ x}{x}$ dx = $\int$ t dt =$\frac{t^2}{2}$
After re-substituting the values, we have
$\frac{(log\ x)^2}{2}$

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