FTC with a two-variable function

• e^(i Pi)+1=0
In summary, differentiation under the integral sign is a mathematical technique for evaluating integrals involving a variable limit of integration. It involves differentiating the integral with respect to a variable, and then integrating the resulting expression. This process can lead to the appearance of additional terms, which can be evaluated using the fundamental theorem of calculus.
e^(i Pi)+1=0
https://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign

I don't understand where the last term comes from, the one that's an integral of a partial derivative. When I solve it using the FTC I get the same answer minus that term.

If I differentiate first then integrate I get that term but then none of the others.

e^(i Pi)+1=0 said:
https://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign

I don't understand where the last term comes from, the one that's an integral of a partial derivative. When I solve it using the FTC I get the same answer minus that term.

If I differentiate first then integrate I get that term but then none of the others.

Work it out for yourself from first principles: if
$$F(x) = \int_{a(x)}^{b(x)} f(t,x) \, dt,$$
then
$$\frac{d}{dx} F(x) = \lim_{h \to 0} \frac{F(x+h) - F(x)}{h}$$
Work out the value of the numerator for small ##h > 0##; you will see that in general it involves several types of terms, and these give limits that are the terms in the final result you want.

1. What is the purpose of using a two-variable function in FTC?

The Fundamental Theorem of Calculus (FTC) with a two-variable function allows us to find the exact value of a definite integral by taking into account the changing rate of the function with respect to two different variables.

2. How do you find the derivative of a two-variable function using FTC?

To find the derivative of a two-variable function using FTC, we use the Chain Rule and differentiate the function with respect to both variables separately, and then multiply the results together.

3. What is the difference between FTC with a two-variable function and single-variable FTC?

In single-variable FTC, we only consider the changing rate of a function with respect to one variable. In FTC with a two-variable function, we consider the changing rate of a function with respect to two different variables, making the integration more accurate.

4. Can FTC with a two-variable function be applied to all types of functions?

Yes, FTC with a two-variable function can be applied to all types of continuous functions, as long as the function has two independent variables. It is a generalization of the single-variable FTC.

5. How is FTC with a two-variable function useful in real-world applications?

FTC with a two-variable function is useful in real-world applications, such as economics and physics, where functions have multiple independent variables. It allows us to accurately calculate rates of change and predict future outcomes.

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