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Calculus

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

Some essential topics are mentioned below and these are thoroughly covered by our online program:

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

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Calculus is the branch of mathematics used to study any phenomena involving change. TutorVista designs suitable online sessions for Math topics. Experienced tutors are available 24/7 and they assist students in solving each calculus problem accurately. The tutors help in understanding each calculus topic and improve your score in exams. Students can choose regular homework help and free question banks are also available online. Get instant online help with experienced tutors and brush up your subject knowledge before exams.

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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
$\frac{(log\ x)^2}{2}$

 More topics in Calculus Rate of Change Derivatives Differentiation Integration Continuity Discontinuity Calculus Homework Help Cauchyâ€“Riemann Equations Entire Function Removable Singularity Methods of Integration Calculus Examples
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