Veerendra

1. What is the application of double integral?

Double integrals are mainly used to calculate the area of a region in the graph, the volume of surface, and the average value of a function of two variables over a rectangular region.

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2. Does order matter in double integral?

In some cases, the order of integral is not a matter. You can change the order of integration. But you need to rewrite the iterated integral.

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3. How do you solve the double integral of a function?

To calculate double integral of any function, you need to compute integral with respect to y, this removes x variable. Then, calculate integral with respect to y.

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4. Can you split double integral?

According to Fubini’s theorem, we can split up double integrals into iterated integrals.

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5. What is the integral used for?

Generally, integrals are used for calculating the area of a two dimensional region, and volume of a 3 dimensional object that is having curved boundary.

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Follow these simple steps to solve any function integration using taylor series. These steps are useful for you to get a clear idea on the concept.

• Let us consider any function to find the integration, upper and lower limits of integration.
• Taylor Series formula for a function f is f(x) = f(a) + f'(a)(x-a) + f”(a)/2 (x-a)² + f”'(a)/3! (x-a)³ + . . . + f⁽ⁿ⁾(a)/n! (x-a)ⁿ
• Where, a is the centre.
• First of all find the derivative of given function and evaluate the derivative at the given point.
• Substitute the obtained values in the above formula to get a polynomial.
• Finally, simplify your polynomial expression to get the final answer.

Example

Question: Calculate Taylor Series of sin(x) up to n=5, centre is 1?

Solution:

Given function is f(x) = sin(x)

n = 0 to 5, centre = 1

Taylor Series formula is f(x) = f(a) + f¹(a)(x-a) + f²(a)/2 (x-a)² + f³(a)/3! (x-a)³ + . . . + f⁽ⁿ⁾(a)/n! (x-a)ⁿ

By using the above values f(x) will become

f(x) = f(0) + f¹(1)(x-1) + f²(2)/2 (x-2)² + f³(3)/3! (x-1)³ + f⁴(4)/4! (x-1)⁴ + f⁵(5)/5! (x-1)⁵

So, calculate the derivative and evaluate them at the given point to get the result into the given formula.

f⁰(x) = f(x) = sin(x)

f(0) = sin(0) = 0

First derivative of f(x) is f¹(x) = [f⁰(x)]’

= [sin(x)]’

= cos(x)

[f(0)]’ = cos(0) = 1

Second Derivative: f²(x) = [f¹(x)]’ = [cos(x)]’ = -sin(x)

(f(0))′′= 0

Evaluate third derivative (f(0))”’ = -cos(0) = -1

Fourth derivative f⁴(x) = [f³(x)]’ = [-cos(x)]’ = sin(x)

(f(0))”” = sin(0) = 0

Fifth derivative: f⁵(x) = [f⁴(x)]’ = [sin(x)]’ = cos(x)

(f(0))””’ = cos(0) = 1

Substitute these values in the taylor formula

f(x) ≈ 0 + 1 (x-1) + 0/2 (x-2)² + (-1)/6 (x-1)³ + 0/4! (x-1)⁴ + 1/5! (x-1)⁵

≈ (x-1) – 1/6 (x-1)³ + 1/120 (x-1)⁵

sin(x) ≈ (x-1) – 1/6 (x-1)³ + 1/120 (x-1)⁵

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1. How do you use Taylor Series Calculator to solve the functions?

The step by step process to use this calculator is provided here. Enter the function, limit and center in the respective input fields. Hit on the calculate button to get the series in the output window.

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2. Solve x²+2x with centre 3 using taylor series?

Given fuunction is x²+2x

Taylor Series formula is f(x) = f(a) + f'(a)(x-a) + f”(a)/2 (x-a)² + f”'(a)/6 (x-a)³

f(3) = (3)²+2(3) = 15

f'(x) = 2x+2

f'(3) = 6+2=8

f”(x) = 2

f”'(x) = 0

Replace the obtained values in the formula

f(x) = 15+ 8(x-3) + 2(x-3)² + 0(x-3)³

x²+2x = 15 + 8(x-3) + 2(x-3)²

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3. What is meant by Taylor Series?

Taylor Series is defined as the expression of the function as the infinite series, in which the terms are expressed for the value of functions derivative at a single point. It is used to draw the conclusion of what a function looks like.

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4. What is Taylor series method?

Taylor series method is one of the earliest analytic numeric algorithms to find approximate solution of initial value problems for ordinary differential equations.

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Below mentioned are the some important derivate rules that are used while solving the derivative of any function. Have a look at them.

1. d/dx (a) = 0 (where a is a constant)
2. d/dx (x) = 1
3. d/dx (xⁿ) = nx^n-1 [power rule]
4. d/dx (e^x) = e^x [exponent rule]
5. d/dx (log x) = 1/x
6. d/dx (a^x) = a^x logx
7. d/dx (f+g) = d/dx (f) + d/dx (g)
8. d/dx (f-g) = d/dx (f) – d/dx (g)
9. d/dx (ay) = a dy/dx
10. (f.g)’ = f’g + g’f [product rule]
11. (f/g)’ = (f’g – g’f) / (g²) [quotient rule]
12. d/dx (f(g(x) = f'(g(x))g'(x) [chain rule]
13. For Trigonometric Functions:
    • d/dx sin(x) = cos(x)
    • d/dx cos(x) = -sin(x)
    • d/dx tan(x) = sec²(x)
    • d/dx cot(x) = -cosec²(x)

Example

Question: Solve derivative of 6 / √z³ + 1 / (8z⁴) – 1 / (3z¹⁰)

Solution:

Given function is 6 / √z³ + 1 / (8z⁴) – 1 / (3z¹⁰)

d/ dz ( 6 / √z³ + 1 / (8z⁴) – 1 / (3z¹⁰)) = ?

= d/ dz ( 6 z^-3/2 + 1/8 (z^-4) – 1/3 (z^-10)

= d/ dz (6 z^-3/2) + d/ dz (1/8 (z^-4)) – d/ dz (1/3 (z^-10))

Apply the power rule i.e d/ dx xⁿ = nx^n-1

= 6 (-3/2) z^{-3/2 – 1} + 1/8 (-4) z^-4-1 -1/3(-10) z^-10-1

= -9z^-5/2 – 1/2 z^-5 + 10/3 z^-11

d/ dz ( 6 / √z³ + 1 / (8z⁴) – 1 / (3z¹⁰)) = -9z^-5/2 – 1/2 z^-5 + 10/3 z^-11

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1. What is derivative formula?

Derivative is the fundamental tool of calculus. The derivative of a function of a real variable measures the sensitivity to change a quantity which is determined by another quantity. Derivative formula is

f¹(x) = lim_△x→0 ([f(x) + △x) – f(x)] / △x

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2. What is the purpose of derivatives?

Derivatives are the financial contracts whose value is linked to the value of an underlying asset.

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3. How do you find the derivative on a calculator?

Enter the input function in the calculator and hit on the calculate button which is provided next to that input box to get the output instantly.

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4. Find the derivative of f(x). Where f(x) = 6x³ – 9x + 4?

f'(x) = 6.3 x^3-1 -9(1) + 0

= 18 x² -9

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