(a) verify that G is an antiderivative of f, (b) find all antiderivatives of f, and (c) sketch t

step 1 When they give you a piece like G of X is equal to 2X, and they're trying to verify that that is

(a) verify that G is an antiderivative of f, (b) find all...

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Key Concepts

Antiderivative

An antiderivative of a function f(x) is a function F(x) such that the derivative of F(x) equals f(x). This concept forms the basis for integration in calculus, as finding an antiderivative is essentially the process of determining an original function from its rate of change.

Constant of Integration

When determining the general antiderivative (or indefinite integral) of a function, an arbitrary constant is added to account for all possible vertical shifts. This is because the derivative of a constant is zero, meaning that any two antiderivatives of the same function differ by a constant.

Verification via Differentiation

Verifying that a given function is an antiderivative involves differentiating the function and checking that the result matches the original function. This process ensures the correctness of the antiderivative and reinforces the relationship between differentiation and integration.

Graphical Interpretation of Antiderivatives

The entire family of antiderivatives of a function is represented by a set of curves that are vertical shifts of one another. Graphing these curves illustrates the effect of the constant of integration, showing that while the slopes (derivatives) of the functions are the same, their y-intercepts vary.

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