The sum of the derivative and integral of a constant equals what? Explain in great detail along with mathematical expressions
Let's start by considering a constant function, f(x) = c, where c is a constant. The derivative of a constant function is always zero, because the value of the function does not change as x varies.
Mathematically, we have:
f(x) = c
f'(x) = 0 (since the derivative of a constant is zero)
Now, let's find the integral of f(x) over an interval [a, b]. The integral of a constant function is computed by multiplying the constant by the width of the interval.
∫[a, b] f(x) dx = ∫[a, b] c dx = c∫[a, b] dx
Integrating dx simply adds the variable x, evaluated at the limits of integration a and b:
∫[a, b] dx = (x)|[a, b] = (b - a)
Therefore:
∫[a, b] f(x) dx = c∫[a, b] dx = c(b - a)
Now, let's sum the derivative and integral of the constant function:
f'(x) + ∫[a, b] f(x) dx = 0 + c(b - a) = c(b - a)
So, the sum of the derivative and integral of a constant function f(x) = c over an interval [a, b] is equal to c times the width of the interval (b - a).
In simpler terms, the derivative of a constant function is always zero, and the integral of a constant function is equal to the constant times the width of the interval. Hence, when you sum them up, you get the same result as the integral, which is c times the width of the interval.
To better understand and explain the sum of the derivative and integral of a constant, let's start by defining the functions involved and then applying the mathematical operations.
First, let's consider a constant function, denoted by f(x) = C, where C is a constant value. This function remains the same for all values of x.
Next, we will take the derivative of this function. The derivative represents the rate of change of a function at any given point. By taking the derivative of a constant function, we can see that the result is always zero. The derivative of f(x) = C is denoted by f'(x) = 0. This means that the derivative of a constant function is a zero function, which has a constant value of zero for all x.
On the other hand, when we integrate a constant function, we are finding the area under the curve of a function over a given interval. In the case of a constant function, the integral is straightforward. The integral of f(x) = C with respect to x is denoted by ∫ f(x) dx = Cx + K, where K is the constant of integration. Note that the integral of a constant function results in a linear function, where the coefficient of x is the constant value.
Now, let's consider the sum of the derivative and integral of a constant function:
∫ f(x) dx + f'(x)
Plugging in the appropriate expressions for the derivative and integral, we have:
∫ f(x) dx + f'(x) = ∫ C dx + 0
Integrating the constant function, we get:
= Cx + K + 0
Simplifying, we have:
= Cx + K
So, the sum of the derivative and integral of a constant function is still a linear function, represented by Cx + K. The result does not depend on x, and the constant value C is the coefficient of x, while K remains constant.
In summary, when we calculate the sum of the derivative and integral of a constant function, the result is another linear function with the same constant value as the original function, but with an arbitrary constant of integration added.