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Chapter 4: Problem 19

Find all the antiderivatives of the following functions. Check your work by taking derivatives. $$f(x)=e^{x}$$

Short Answer

Expert verified
Answer: The antiderivative of the function $$f(x) = e^x$$ is $$F(x) = e^x + C$$, where $$C$$ is the constant of integration.

Step by step solution

01

Integrate the function

To find the antiderivative of the function, we will integrate $$f(x) = e^x$$ with respect to x. The integral of this function is: $$\int e^x \, dx$$
02

Find the antiderivative

The integral of $$e^x$$ is itself, so the antiderivative of $$f(x) = e^x$$ is: $$F(x) = e^x + C$$ Here, $$F(x)$$ is the antiderivative of $$f(x)$$, and $$C$$ denotes the constant of integration (because the derivative of a constant is always 0).
03

Check the solution by taking the derivative

To confirm that our solution is correct, we will check that the derivative of our antiderivative $$F(x)$$ is equal to the original function $$f(x)$$. Using the power rule, we find the derivative of $$F(x) = e^x + C$$: $$F'(x) = \frac{d}{dx} (e^x + C) = e^x + 0$$
04

Compare the result with the original function

Our derivative $$F'(x) = e^x$$ matches the original function $$f(x) = e^x$$, which confirms that our solution is correct. Therefore, the antiderivatives of the function $$f(x) = e^x$$ are: $$F(x) = e^x + C$$

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Most popular questions from this chapter

a. For what values of \(b>0\) does \(b^{x}\) grow faster than \(e^{x}\) as \(x \rightarrow \infty ?\) b. Compare the growth rates of \(e^{x}\) and \(e^{a x}\) as \(x \rightarrow \infty,\) for \(a>0\).

Use the identities \(\sin ^{2} x=(1-\cos 2 x) / 2\) and \(\cos ^{2} x=(1+\cos 2 x) / 2\) to find \(\int \sin ^{2} x d x\) and \(\int \cos ^{2} x d x\).

Locate the critical points of the following functions and use the Second Derivative Test to determine whether they correspond to local maxima, local minima, or neither. $$h(x)=(x+a)^{4}, a \text { constant }$$

Differentials Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\) $$f(x)=e^{2 x}$$

Consider the general cubic polynomial \(f(x)=x^{3}+a x^{2}+b x+c,\) where \(a, b,\) and \(c\) are real numbers. a. Prove that \(f\) has exactly one local maximum and one local minimum provided that \(a^{2}>3 b\) b. Prove that \(f\) has no extreme values if \(a^{2}<3 b\)
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