Problem 18 - Calculus Early Transcendentals

Question

Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. $$H(z)=-6 z^{-7}$$

Step-by-Step Solution

Answer

Question: Find all the antiderivatives of the function H(z) = -6z^(-7). Answer: The antiderivatives of the function H(z) = -6z^(-7) are given by G(z) = z^(-6) + C, where C represents the constant of integration.

Step 1: Identify the power of z in the function

In the given function, H(z) = -6z^(-7), the power of z is -7.

Step 2: Find the antiderivative using the power rule for integration

According to the power rule, the antiderivative of z^n is z^(n+1)/(n+1) + C. In our case, n = -7. Thus, the antiderivative of H(z) is: $$\int H(z) dz = \int -6z^{-7} dz = -6 \int z^{-7} dz$$ Now we apply the power rule: $$-6\left[\frac{z^{-7+1}}{-7+1} + C\right] = -6\left[\frac{z^{-6}}{-6} + C\right]$$ Finally, simplify the expression: $$G(z) = z^{-6} + C$$

Step 3: Check the result by taking the derivative

To make sure that we have found the correct antiderivative, we need to take the derivative of G(z) and check if we get the original function H(z): $$G'(z) = \frac{d}{dz}\left(z^{-6} + C\right)$$ Applying the power rule for derivatives, we get: $$G'(z) = -6z^{-7}$$ Since G'(z) = H(z), our antiderivative is correct. So, the antiderivatives of the given function are: $$G(z) = z^{-6} + C$$

Key Concepts

Power rule for integration

The power rule for integration is a fundamental technique used in calculus to find an antiderivative, which is the opposite process of finding a derivative. This rule applies to functions of the form \( z^n \), where \( n \) is any real number except \( -1 \). The power rule states that the antiderivative of \( z^n \) can be found by increasing the exponent by one and dividing by the new exponent.

• Add 1 to the exponent: If the function is \( z^n \), then the new exponent becomes \( n+1 \).
• Divide by the new exponent: The function's coefficient is divided by this new exponent \( n+1 \).
• Don't forget the constant of integration: Always add \( C \) at the end, because an antiderivative is not unique. \( C \) represents any constant, accounting for the family of functions that differ by a constant.

For example, in the exercise, \( H(z) = -6z^{-7} \). Applying the power rule, the exponent \( -7 \) is increased by 1 to \( -6 \), and then we divide by \( -6 \), yielding the antiderivative \( G(z) = z^{-6} + C \). This simplifies the integration process and helps verify the solution's correctness.

Definite integral

A definite integral is an integral that is evaluated over a specific interval, providing the exact value that represents the accumulated area under the curve of a function between two limits. Unlike an indefinite integral, which yields a general function plus a constant, a definite integral results in a numerical value.

• Limits of integration: Definite integrals require upper and lower limits to define the region over which integration occurs.
• Net area: It measures the net area under the curve, considering areas above the x-axis as positive and below as negative.
• No constant of integration: Since it results in a specific value, there is no need for \( C \).

For instance, if you wanted to find the total change represented by \( H(z) = -6z^{-7} \) over an interval from \( z = a \) to \( z = b \), you would evaluate the definite integral, using the antiderivative found, turning it into an exact numerical representation of the change rather than just a function expression.

Indefinite integral

An indefinite integral refers to finding an antiderivative of a function, which includes a family of functions rather than a single value. It is represented without specific limits, indicating a general form of the original function that, when differentiated, reproduces the original function.

• General solution: The indefinite integral embodies a range of possible functions because of the constant of integration, \( C \).
• Symbolized by \( \int f(z) \, dz \): It is written with the integral symbol and no limits, stressing the non-specificity regarding the interval or boundaries.
• Added constant \( C \): This crucial constant differentiates it from a definite solution, allowing for the multiple solutions that differ only by a constant.

For \( H(z) = -6z^{-7} \), finding the indefinite integral leads us to \( G(z) = z^{-6} + C \), capturing every possible antiderivative version of the function. The indefinite integral is essential in providing a broad understanding of function behavior across variable conditions.