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

(a) verify that $G$ is an antiderivative of $f$, (b) find all antiderivatives of $f$, and $(c)$ sketch the graphs of a few of the family of antiderivatives found in part (b).
$$G(x)=e^{x} ; f(x)=e^{x}$$

Key Concepts

Antiderivative Verification

An antiderivative of a function is another function whose derivative equals the original function. Verifying that a given function is an antiderivative involves differentiating it and checking that the result matches the original function. This concept is fundamental in calculus, linking differentiation and integration as inverse processes.

General Antiderivative and Constant of Integration

When finding all antiderivatives of a given function, one must account for the constant of integration. This constant arises because the derivative of a constant is zero, meaning that any two antiderivatives of the same function can differ by an arbitrary constant. The general antiderivative is typically expressed as a specific antiderivative plus an arbitrary constant.

Graphical Representation of Antiderivatives

Graphing a family of antiderivatives illustrates a set of curves that are vertically shifted versions of each other. Each curve in the family represents an antiderivative that differs by a constant, demonstrating the idea that while the overall shape of the graph remains the same, the curves are translated vertically based on different values of the constant of integration.

Transcript

00:01 All right, we are asked to verify that this function g is the anti -derivative of f.

00:08 Well, to verify that something's the anti -derivative, what you want to do is take the derivative.

00:15 We learned in an earlier section that the derivative of e to the x is e to the x, which verifies, since that is the same thing, that it is the correct anti -derivative.

00:28 So then in part b, what they want you to come up with is any function.

00:32 I'm going to call it capital f just to change it from lowercase f.

00:38 And the key to this is understanding that it's going to be the same thing as g of x, so the correct antiderivative.

00:45 But then you have to add some constant to this.

00:49 And the reason why we need a constant is we know that the derivative of a constant is always zero, but we don't write plus zero up here.

00:58 And basically what it breaks down to is we can shift up and down however many times we want, but the slope remains the same throughout, which brings me to the part c when we're graphing all of these, because we could just graph the graph of the exponential function, which passes through at 0 .1, it has a horizontal asymptote at y equals 0.

01:26 And the slope of this curve is e to the x throughout.

01:31 But let's shift it up a little bit higher.

01:33 I'll just go to like you have a plus 2.

01:36 So it goes up to 0 .3...

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