[TOC]

### Question 1: Evaluate the following function expressions

#### (a)
Evaluate $f(5) - f(1)$ for the function:

$$
f(x) = 7x^4 - 4x^2 + 9x
$$

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<summary> Example: </summary>

### Problem:

Evaluate $f(6) - f(2)$ for the function:

$f(x) = 5x^5 - 3x^3 + 6x^2 - 8x$

### Solution:

First, calculate f(6):

\begin{align*}
f(6) &= 5(6)^5 - 3(6)^3 + 6(6)^2 - 8(6) \\
&= 5(7776) - 3(216) + 6(36) - 8(6) \\
&= 38880 - 648 + 216 - 48 \\
&= 38240
\end{align*}

Now, calculate f(2):

\begin{align*}
f(2) &= 5(2)^5 - 3(2)^3 + 6(2)^2 - 8(2) \\
&= 5(32) - 3(8) + 6(4) - 8(2) \\
&= 160 - 24 + 24 - 16 \\
&= 144
\end{align*}

Finally, calculate $f(6) - f(2)$:

\begin{align*}
f(6) - f(2) &= 38240 - 144 \\
&= 38096
\end{align*}

Thus, the value of $f(6) - f(2)$ is:

**`38096`**

:::

#### (b)
Evaluate $g(3) - g(-1)$ for the function:

$$
g(x) = -2x^3 + 5x^2 - 3
$$

::: spoiler

<summary> Example: </summary>

We are tasked with evaluating $g(2) - g(-1)$ for the function:

$$
g(x) = -3x^4 + 4x^3 - 4
$$

### Step 1: Compute $g(2)$

We substitute $x = 2$ into the expression for $g(x)$:

$$
g(2) = -3(2)^4 + 4(2)^3 - 4
$$

First, calculate the powers of 2:

$$
2^4 = 16 \quad \text{and} \quad 2^3 = 8
$$

Now substitute these values back into the equation:

$$
g(2) = -3(16) + 4(8) - 4
$$

Perform the multiplications:

$$
g(2) = -48 + 32 - 4
$$

Now, sum the terms:

$$
g(2) = -48 + 32 = -16
$$
$$
g(2) = -16 - 4 = -20
$$

Thus, $g(2) = -20$.

### Step 2: Compute $g(-1)$

Now substitute $x = -1$ into the expression for $g(x)$:

$$
g(-1) = -3(-1)^4 + 4(-1)^3 - 4
$$

First, calculate the powers of $-1$:

$$
(-1)^4 = 1 \quad \text{and} \quad (-1)^3 = -1
$$

Now substitute these values back into the equation:

$$
g(-1) = -3(1) + 4(-1) - 4
$$

Perform the multiplications:

$$
g(-1) = -3 - 4 - 4
$$

Now, sum the terms:

$$
g(-1) = -3 - 4 = -7
$$
$$
g(-1) = -7 - 4 = -11
$$

Thus, $g(-1) = -11$.

### Step 3: Compute $g(2) - g(-1)$

Now, subtract $g(-1)$ from $g(2)$:

$$
g(2) - g(-1) = -20 - (-11)
$$

Simplify the expression:

$$
g(2) - g(-1) = -20 + 11 = -9
$$

### Final Answer:

$$
g(2) - g(-1) = -9
$$

:::

#### (c)
Evaluate $h(10) - h(-10)$ for the function:

$$
h(x) = 4x^5 - 2x^4 + 3x - 1
$$

::: spoiler

<summary> Example: </summary>

We are tasked with evaluating $h(9) - h(-9)$ for the function:

$$
h(x) = 2x^3 - 3x^2 + 4x - 5
$$

### Step 1: Compute $h(9)$

We substitute $x = 9$ into the expression for $h(x)$:

$$
h(9) = 2(9)^3 - 3(9)^2 + 4(9) - 5
$$

First, calculate the powers of 9:

$$
9^3 = 729 \quad \text{and} \quad 9^2 = 81
$$

Now substitute these values back into the equation:

$$
h(9) = 2(729) - 3(81) + 4(9) - 5
$$

Perform the multiplications:

$$
h(9) = 1458 - 243 + 36 - 5
$$

Now, sum the terms:

$$
h(9) = 1458 - 243 = 1215
$$
$$
h(9) = 1215 + 36 = 1251
$$
$$
h(9) = 1251 - 5 = 1246
$$

Thus, $h(9) = 1246$.

### Step 2: Compute $h(-9)$

Now substitute $x = -9$ into the expression for $h(x)$:

$$
h(-9) = 2(-9)^3 - 3(-9)^2 + 4(-9) - 5
$$

First, calculate the powers of $-9$:

$$
(-9)^3 = -729 \quad \text{and} \quad (-9)^2 = 81
$$

Now substitute these values back into the equation:

$$
h(-9) = 2(-729) - 3(81) + 4(-9) - 5
$$

Perform the multiplications:

$$
h(-9) = -1458 - 243 - 36 - 5
$$

Now, sum the terms:

$$
h(-9) = -1458 - 243 = -1701
$$
$$
h(-9) = -1701 - 36 = -1737
$$
$$
h(-9) = -1737 - 5 = -1742
$$

Thus, $h(-9) = -1742$.

### Step 3: Compute $h(9) - h(-9)$

Now, subtract $h(-9)$ from $h(9)$:

$$
h(9) - h(-9) = 1246 - (-1742)
$$

Simplify the expression:

$$
h(9) - h(-9) = 1246 + 1742 = 2988
$$

### Final Answer:

$$
h(9) - h(-9) = 2988
$$

:::

---

### Question 2: Find the following derivatives

#### (a)
Find $f'(x)$ for the function:

$$
f(x) = 7x^4 - 4x^2 + 9x
$$

::: spoiler

<summary> Example: </summary>

We are tasked with finding the derivative, $f'(x)$, of the function:

$$
f(x) = 6x^5 - 3x^3 + 7x
$$

### Step 1: Recall the Power Rule

To differentiate the function, we will apply the **power rule** for derivatives. The power rule states:

$$
\frac{d}{dx} \left( x^n \right) = nx^{n-1}
$$

We will apply this rule to each term of the function.

### Step 2: Differentiate Each Term

The function $f(x)$ has three terms: $6x^5$, $-3x^3$, and $7x$. We will differentiate each of them separately.

#### 1. Differentiate $6x^5$
Using the power rule with $n = 5$, we differentiate $6x^5$:

$$
\frac{d}{dx} \left( 6x^5 \right) = 6 \cdot 5x^{5-1} = 30x^4
$$

#### 2. Differentiate $-3x^3$
Using the power rule with $n = 3$, we differentiate $-3x^3$:

$$
\frac{d}{dx} \left( -3x^3 \right) = -3 \cdot 3x^{3-1} = -9x^2
$$

#### 3. Differentiate $7x$
For the term $7x$, note that this is a linear term ($x^1$), so using the power rule with $n = 1$:

$$
\frac{d}{dx} \left( 7x \right) = 7 \cdot 1x^{1-1} = 7
$$

### Step 3: Combine the Results

Now, we combine the derivatives of each term:

$$
f'(x) = 30x^4 - 9x^2 + 7
$$

### Final Answer:

The derivative of the function is:

$$
f'(x) = 30x^4 - 9x^2 + 7
$$

:::

#### (b)
Find $g'(x)$ for the function:

$$
g(x) = -2x^3 + 5x^2 - 3
$$

::: spoiler

<summary> Example: </summary>

We are tasked with finding the derivative, $g'(x)$, of the function:

$$
g(x) = -4x^3 + 6x^2 - 5
$$

### Step 1: Recall the Power Rule

To differentiate the function, we will apply the **power rule** for derivatives. The power rule states:

$$
\frac{d}{dx} \left( x^n \right) = nx^{n-1}
$$

We will apply this rule to each term of the function.

### Step 2: Differentiate Each Term

The function $g(x)$ has three terms: $-4x^3$, $6x^2$, and $-5$. We will differentiate each of them separately.

#### 1. Differentiate $-4x^3$
Using the power rule with $n = 3$, we differentiate $-4x^3$:

$$
\frac{d}{dx} \left( -4x^3 \right) = -4 \cdot 3x^{3-1} = -12x^2
$$

#### 2. Differentiate $6x^2$
Using the power rule with $n = 2$, we differentiate $6x^2$:

$$
\frac{d}{dx} \left( 6x^2 \right) = 6 \cdot 2x^{2-1} = 12x
$$

#### 3. Differentiate $-5$
Since $-5$ is a constant, the derivative of a constant is 0:

$$
\frac{d}{dx} \left( -5 \right) = 0
$$

### Step 3: Combine the Results

Now, we combine the derivatives of each term:

$$
g'(x) = -12x^2 + 12x + 0
$$

Simplifying, we have:

$$
g'(x) = -12x^2 + 12x
$$

### Final Answer:

The derivative of the function is:

$$
g'(x) = -12x^2 + 12x
$$

:::

#### (c)
Find $h'(x)$ for the function:

$$
h(x) = 4x^5 - 2x^4 + 3x - 1
$$

::: spoiler

<summary> Example: </summary>

We are tasked with finding the derivative, $h'(x)$, of the function:

$$
h(x) = 5x^6 - 3x^4 + 4x - 2
$$

### Step 1: Recall the Power Rule

To differentiate the function, we will apply the **power rule** for derivatives. The power rule states:

$$
\frac{d}{dx} \left( x^n \right) = nx^{n-1}
$$

We will apply this rule to each term of the function.

### Step 2: Differentiate Each Term

The function $h(x)$ has four terms: $5x^6$, $-3x^4$, $4x$, and $-2$. We will differentiate each of them separately.

#### 1. Differentiate $5x^6$
Using the power rule with $n = 6$, we differentiate $5x^6$:

$$
\frac{d}{dx} \left( 5x^6 \right) = 5 \cdot 6x^{6-1} = 30x^5
$$

#### 2. Differentiate $-3x^4$
Using the power rule with $n = 4$, we differentiate $-3x^4$:

$$
\frac{d}{dx} \left( -3x^4 \right) = -3 \cdot 4x^{4-1} = -12x^3
$$

#### 3. Differentiate $4x$
For the term $4x$, note that this is a linear term ($x^1$), so using the power rule with $n = 1$:

$$
\frac{d}{dx} \left( 4x \right) = 4 \cdot 1x^{1-1} = 4
$$

#### 4. Differentiate $-2$
Since $-2$ is a constant, the derivative of a constant is 0:

$$
\frac{d}{dx} \left( -2 \right) = 0
$$

### Step 3: Combine the Results

Now, we combine the derivatives of each term:

$$
h'(x) = 30x^5 - 12x^3 + 4 + 0
$$

Simplifying, we have:

$$
h'(x) = 30x^5 - 12x^3 + 4
$$

### Final Answer:

The derivative of the function is:

$$
h'(x) = 30x^5 - 12x^3 + 4
$$

:::

### Question 3: Find each indefinite integral

#### (a)
Evaluate the indefinite integral:

$$
\int x^5 \, dx =
$$

::: spoiler

<summary> Example: </summary>

We are tasked with evaluating the indefinite integral:

$$
\int x^6 \, dx
$$

### Step 1: Recall the Power Rule for Integration

To evaluate this integral, we will apply the **power rule for integration**. The power rule states:

$$
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for} \ n \neq -1
$$

Here, $C$ is the constant of integration, which must always be included when evaluating indefinite integrals.

### Step 2: Apply the Power Rule

In this case, we have $n = 6$. Using the power rule, we increase the exponent by 1 and divide by the new exponent:

$$
\int x^6 \, dx = \frac{x^{6+1}}{6+1} + C = \frac{x^7}{7} + C
$$

### Final Answer:

Thus, the indefinite integral of $x^6$ is:

$$
\int x^6 \, dx = \frac{x^7}{7} + C
$$

:::

#### (b)
Evaluate the indefinite integral:

$$
\int 15x^2 \, dx =
$$

::: spoiler

<summary> Example: </summary>

We are tasked with evaluating the indefinite integral:

$$
\int 16x^3 \, dx
$$

### Step 1: Recall the Power Rule for Integration

To evaluate this integral, we will apply the **power rule for integration**. The power rule states:

$$
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for} \ n \neq -1
$$

Here, $C$ is the constant of integration, which must always be included when evaluating indefinite integrals.

### Step 2: Apply the Power Rule

In this case, we have $n = 3$ and a constant factor of 16. The constant can be factored out of the integral, so we first rewrite the integral:

$$
\int 16x^3 \, dx = 16 \int x^3 \, dx
$$

Now, using the power rule for $x^3$, we increase the exponent by 1 and divide by the new exponent:

$$
\int x^3 \, dx = \frac{x^{3+1}}{3+1} = \frac{x^4}{4}
$$

Multiply this result by 16:

$$
16 \cdot \frac{x^4}{4} = 4x^4
$$

### Step 3: Include the Constant of Integration

Don't forget to add the constant of integration, $C$:

$$
4x^4 + C
$$

### Final Answer:

Thus, the indefinite integral of $16x^3$ is:

$$
\int 16x^3 \, dx = 4x^4 + C
$$

:::

#### (c)
Evaluate the indefinite integral:

$$
\int x^{-4} \, dx =
$$

::: spoiler

<summary> Example: </summary>

We are tasked with evaluating the indefinite integral:

$$
\int x^{-5} \, dx
$$

### Step 1: Recall the Power Rule for Integration

To evaluate this integral, we will apply the **power rule for integration**. The power rule states:

$$
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for} \ n \neq -1
$$

Here, $C$ is the constant of integration, which must always be included when evaluating indefinite integrals.

### Step 2: Apply the Power Rule

In this case, we have $n = -5$. Using the power rule, we increase the exponent by 1 and divide by the new exponent:

$$
\int x^{-5} \, dx = \frac{x^{-5+1}}{-5+1} = \frac{x^{-4}}{-4}
$$

### Step 3: Simplify and Include the Constant of Integration

We simplify the result as:

$$
\frac{x^{-4}}{-4} = -\frac{1}{4}x^{-4}
$$

Don't forget to add the constant of integration, $C$:

$$
-\frac{1}{4}x^{-4} + C
$$

### Final Answer:

Thus, the indefinite integral of $x^{-5}$ is:

$$
\int x^{-5} \, dx = -\frac{1}{4}x^{-4} + C
$$

:::

#### (d)
Evaluate the indefinite integral:

$$
\int 8x^{1/3} \, dx =
$$

::: spoiler

<summary> Example: </summary>

We are tasked with evaluating the indefinite integral:

$$
\int 20x^{1/4} \, dx
$$

### Step 1: Recall the Power Rule for Integration

To evaluate this integral, we will apply the **power rule for integration**. The power rule states:

$$
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for} \ n \neq -1
$$

Here, $C$ is the constant of integration, which must always be included when evaluating indefinite integrals.

### Step 2: Apply the Power Rule

In this case, we have $n = \frac{1}{4}$ and a constant factor of 20. The constant can be factored out of the integral, so we first rewrite the integral:

$$
\int 20x^{1/4} \, dx = 20 \int x^{1/4} \, dx
$$

Now, using the power rule for $x^{1/4}$, we increase the exponent by 1 and divide by the new exponent:

$$
\int x^{1/4} \, dx = \frac{x^{1/4 + 1}}{1/4 + 1} = \frac{x^{5/4}}{5/4}
$$

### Step 3: Simplify the Expression

Simplify the fraction $\frac{1}{5/4}$ by multiplying by the reciprocal:

$$
\frac{x^{5/4}}{5/4} = \frac{4x^{5/4}}{5}
$$

Now, multiply by the constant 20:

$$
20 \cdot \frac{4x^{5/4}}{5} = \frac{80x^{5/4}}{5} = 16x^{5/4}
$$

### Step 4: Include the Constant of Integration

Don't forget to add the constant of integration, $C$:

$$
16x^{5/4} + C
$$

### Final Answer:

Thus, the indefinite integral of $20x^{1/4}$ is:

$$
\int 20x^{1/4} \, dx = 16x^{5/4} + C
$$

:::

### Question 4: Find each indefinite integral

#### (a)
Evaluate the indefinite integral:

$$
\int \frac{1 - y^2}{3y} \, dy =
$$

::: spoiler

<summary> Example: </summary>

We are tasked with evaluating the indefinite integral:

$$
\int \frac{2y - y^3}{3y^2} \, dy
$$

### Step 1: Simplify the Expression

Before applying the power rule for integration, we simplify the expression by dividing each term in the numerator by $3y^2$:

$$
\frac{2y - y^3}{3y^2} = \frac{2y}{3y^2} - \frac{y^3}{3y^2}
$$

Simplify each term:

$$
\frac{2y}{3y^2} = \frac{2}{3y}, \quad \frac{y^3}{3y^2} = \frac{y}{3}
$$

So the expression becomes:

$$
\frac{2y - y^3}{3y^2} = \frac{2}{3y} - \frac{y}{3}
$$

Thus, the integral becomes:

$$
\int \left( \frac{2}{3y} - \frac{y}{3} \right) \, dy
$$

### Step 2: Break the Integral into Two Terms

Now, break the integral into two separate integrals:

$$
\int \frac{2}{3y} \, dy - \int \frac{y}{3} \, dy
$$

### Step 3: Evaluate Each Integral

#### 1. Evaluate $\int \frac{2}{3y} \, dy$

This is a standard logarithmic integral. We rewrite it as:

$$
\frac{2}{3} \int \frac{1}{y} \, dy = \frac{2}{3} \ln|y|
$$

#### 2. Evaluate $\int \frac{y}{3} \, dy$

This is a power rule integral. Using the power rule for $y^1$, we get:

$$
\frac{1}{3} \int y \, dy = \frac{1}{3} \cdot \frac{y^2}{2} = \frac{y^2}{6}
$$

### Step 4: Combine the Results and Add the Constant of Integration

Now, combine the two integrals:

$$
\frac{2}{3} \ln|y| - \frac{y^2}{6} + C
$$

### Final Answer:

Thus, the indefinite integral of $\frac{2y - y^3}{3y^2}$ is:

$$
\int \frac{2y - y^3}{3y^2} \, dy = \frac{2}{3} \ln|y| - \frac{y^2}{6} + C
$$

:::

#### (b)
Evaluate the indefinite integral:

$$
\int \left( 4x^3 + \frac{2}{x^3} \right) \, dx =
$$

::: spoiler

<summary> Example: </summary>

We are tasked with evaluating the indefinite integral:

$$
\int \left( 5x^4 + \frac{9}{x^4} \right) \, dx
$$

### Step 1: Rewrite the Terms

Before applying the power rule, rewrite $\frac{9}{x^4}$ as a power of $x$. We can express it as:

$$
\frac{9}{x^4} = 9x^{-4}
$$

Thus, the integral becomes:

$$
\int \left( 5x^4 + 9x^{-4} \right) \, dx
$$

### Step 2: Break the Integral into Two Terms

Now, break the integral into two separate integrals:

$$
\int 5x^4 \, dx + \int 9x^{-4} \, dx
$$

### Step 3: Apply the Power Rule for Integration

We will now apply the **power rule for integration**, which states:

$$
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for} \ n \neq -1
$$

#### 1. Evaluate $\int 5x^4 \, dx$

Using the power rule with $n = 4$, we increase the exponent by 1 and divide by the new exponent:

$$
\int 5x^4 \, dx = 5 \cdot \frac{x^{4+1}}{4+1} = \frac{5x^5}{5} = x^5
$$

#### 2. Evaluate $\int 9x^{-4} \, dx$

Using the power rule with $n = -4$, we increase the exponent by 1 and divide by the new exponent:

$$
\int 9x^{-4} \, dx = 9 \cdot \frac{x^{-4+1}}{-4+1} = 9 \cdot \frac{x^{-3}}{-3} = -3x^{-3}
$$

### Step 4: Rewrite Without Negative Exponents and Add the Constant of Integration

We rewrite $x^{-3}$ as $\frac{1}{x^3}$ and combine the two integrals:

$$
x^5 - \frac{3}{x^3} + C
$$

### Final Answer:

Thus, the indefinite integral of $\left( 5x^4 + \frac{9}{x^4} \right)$ is:

$$
\int \left( 5x^4 + \frac{9}{x^4} \right) \, dx = x^5 - \frac{3}{x^3} + C
$$

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