Antiderivatives: Videos & Practice Problems

Antiderivatives represent the reverse process of derivatives, allowing us to find a function from its derivative. For example, the antiderivative of \(2x\) is \(x^2 + c\), where \(c\) is an unknown constant. To find particular antiderivatives, we can use given points to solve for \(c\). For instance, if \(f(2) = 3\), substituting gives \(2^2 + c = 3\), leading to \(c = -1\). Thus, the particular antiderivative is \(x^2 - 1\). This process highlights the importance of understanding both general and specific solutions in calculus.

Antiderivatives

Understanding antiderivatives is essential as it represents the reverse process of taking derivatives. While derivatives provide the rate of change of a function, antiderivatives help us recover the original function from its derivative. This concept is crucial as we progress in calculus.

To illustrate, consider the function \( f(x) = x^2 + 10 \). The derivative, using the power rule, is \( f'(x) = 2x \). To find the antiderivative, we reverse this process. The antiderivative of \( 2x \) is \( F(x) = x^2 + C \), where \( C \) is an arbitrary constant. This constant arises because the derivative of a constant is zero, meaning we cannot determine its value from the derivative alone.

Let’s explore a few examples to solidify this understanding:

In the first example, we find the antiderivative of \( 3x^2 \). Recognizing that the derivative of \( x^3 \) is \( 3x^2 \), we conclude that the antiderivative is \( F(x) = x^3 + C \).

In the second example, we consider the constant function \( 3 \). The antiderivative of a constant is a linear function, so we find \( F(x) = 3x + C \).

For the third example, we look for the antiderivative of \( f(x) = 0 \). The only function whose derivative is zero is a constant function, leading us to \( F(x) = C \).

To verify our results, we can differentiate our antiderivatives. For instance, differentiating \( F(x) = x^3 + C \) yields \( f(x) = 3x^2 \), confirming our solution. Similarly, differentiating \( F(x) = 3x + C \) gives \( f(x) = 3 \), and differentiating \( F(x) = C \) results in \( f(x) = 0 \). Each check aligns with the original functions, validating our antiderivative calculations.

In summary, mastering the process of finding antiderivatives is vital in calculus, as it allows us to work backwards from derivatives to recover original functions. The inclusion of the constant \( C \) is a key aspect of this process, reflecting the inherent uncertainty in determining the original function from its derivative alone.

Find the antiderivative of the following function.
$f\left(x\right)=200$

• A. $F\left(x\right)=0$
• B. $F\left(x\right)=200+C$
• C. $F\left(x\right)=200x$
• D. $F\left(x\right)=200x+C$

Answer: (D)

Find the antiderivative of the following function.
$f\left(x\right)=5x^4$

• A. $F\left(x\right)=25x^5+C$
• B. $F\left(x\right)=x^5+C$
• C. $F\left(x\right)=5x^5+C$
• D. $F\left(x\right)=\frac54x^5+C$

Answer: (B)

Find the antiderivative of the following function.
$f\left(x\right)=10x^9$

• A. $F\left(x\right)=x^{10}+C$
• B. $F\left(x\right)=10x^{10}+C$
• C. $F\left(x\right)=100x^{10}+C$
• D. $F\left(x\right)=\frac{10}{9}x^{10}+C$

Answer: (A)

Finding a Particular Antiderivative

In calculus, finding an antiderivative is the reverse process of differentiation. When we take the antiderivative of a function, we obtain a general form that includes a constant, represented as \( F(x) = f(x) + C \), where \( C \) is an arbitrary constant. This general antiderivative represents a family of functions that differ by vertical shifts on the graph.

To find a particular antiderivative, we need additional information, typically a point on the function. For example, if we are given the function \( f(x) = 2x \), we can determine its general antiderivative by recognizing that the derivative of \( x^2 \) is \( 2x \). Thus, the general antiderivative is \( F(x) = x^2 + C \). If we know that \( F(2) = 3 \), we can substitute \( x = 2 \) into the equation:

\( F(2) = 2^2 + C = 3 \)

This simplifies to:

\( 4 + C = 3 \)

Solving for \( C \) gives us \( C = 3 - 4 = -1 \). Therefore, the particular antiderivative is:

\( F(x) = x^2 - 1 \)

In another example, consider \( f(x) = 3x^2 \). The general antiderivative can be found by recognizing that the derivative of \( x^3 \) is \( 3x^2 \), leading to:

\( F(x) = x^3 + C \)

If we are given that \( F(1) = 5 \), we substitute \( x = 1 \):

\( F(1) = 1^3 + C = 5 \)

This simplifies to:

\( 1 + C = 5 \)

Solving for \( C \) gives \( C = 5 - 1 = 4 \). Thus, the particular antiderivative is:

\( F(x) = x^3 + 4 \)

In summary, to find a particular antiderivative, start by determining the general antiderivative and then use the given point to solve for the constant \( C \). This process allows us to pinpoint a specific function from the infinite family represented by the general antiderivative.

For the following function $f\left(x\right)$, find the antiderivative $F\left(x\right)$ that satisfies the given condition.
$f\left(x\right)=5x^4$; $F\left(0\right)=1$

• A. $F\left(x\right)=5x^5+1$
• B. $F\left(x\right)=x^5+1$
• C. $F\left(x\right)=25x^5+1$
• D. $F\left(x\right)=x^5$

Answer: (B)

For the following function $f\left(x\right)$, find the antiderivative $F\left(x\right)$ that satisfies the given condition.
$f\left(x\right)=100x^{99}$; $F\left(1\right)=101$

• A. $F\left(x\right)=x^{100}+101$
• B. $F\left(x\right)=x^{100}+100$
• C. $F\left(x\right)=100x^{100}+101$
• D. $F\left(x\right)=100x^{100}+1$

Answer: (B)

Here’s what students ask on this topic:

What is an antiderivative and how is it related to derivatives?

An antiderivative is essentially the reverse process of taking a derivative. While a derivative gives the rate of change of a function, an antiderivative allows us to find the original function from its derivative. For example, if the derivative of a function is $f\prime \left(x\right)=2x$, then the antiderivative is $F\left(x\right)=x^2+c$, where $c$ is an unknown constant. This constant arises because the derivative of any constant is zero, so it is lost in the differentiation process. Understanding antiderivatives is crucial in calculus as it forms the basis for integral calculus, which deals with finding areas under curves and solving differential equations.

How do you find the particular antiderivative of a function?

To find the particular antiderivative of a function, you first determine the general antiderivative, which includes an unknown constant $c$. For example, the general antiderivative of $2x$ is $x^2+c$. To find the particular antiderivative, you use a given point on the function, such as $f\left(2\right)=3$. Substitute this point into the general antiderivative to solve for $c$. In this case, $2{\left(}^2+c=3$ leads to $c=-1$. Thus, the particular antiderivative is $x^2-1$. This process is essential for solving problems where specific conditions are given.

Why is the constant $c$ important in antiderivatives?

The constant $c$ in antiderivatives is crucial because it accounts for any constant that was present in the original function before differentiation. When taking a derivative, any constant term becomes zero, so it is not reflected in the derivative. Therefore, when finding an antiderivative, we add $c$ to represent this unknown constant. This constant is important because it allows us to describe a family of functions that differ only by a vertical shift. In practical applications, such as solving differential equations, determining the value of $c$ using initial conditions or boundary values is essential to find the specific solution that fits the given context.

How can you verify if an antiderivative is correct?

To verify if an antiderivative is correct, you can differentiate the antiderivative and check if it matches the original function. For instance, if you have an antiderivative $F\left(x\right)=x^2+c$ for the function $f\left(x\right)=2x$, differentiate $F\left(x\right)$ to get $F\prime \left(x\right)=2x$. If this matches the original function, the antiderivative is correct. This process ensures that the reverse operation of differentiation was performed accurately. It is a useful technique to confirm your work, especially when dealing with complex functions or when solving problems that require precise solutions.

What is the difference between a general and a particular antiderivative?

A general antiderivative includes an unknown constant $c$ and represents a family of functions that differ by a constant. For example, the general antiderivative of $2x$ is $x^2+c$. This constant accounts for any vertical shift in the function. A particular antiderivative, on the other hand, is a specific function obtained by determining the value of $c$ using additional information, such as a point on the function. For instance, if $f\left(2\right)=3$, solving for $c$ gives a particular antiderivative like $x^2-1$. Understanding both types is essential for solving calculus problems accurately.