Calculus Study Guide: Average Velocity, Limits, Asymptotes, and Continuity

Q1. Find the average velocity of $y = x^2 + 9x$ over the interval $[2,7]$.

Background

Topic: Average Velocity (Difference Quotient)

This question tests your understanding of how to compute the average rate of change (velocity) of a function over a given interval, which is foundational for calculus and leads to the concept of derivatives.

Key formula:

$v_{av} = \frac{y(b) - y(a)}{b - a}$

• $y(x)$: The function value at $x$

• $a, b$: The endpoints of the interval

Step-by-Step Guidance

1. Calculate $y(7)$ by plugging $x = 7$ into the function: $y(7) = 7^2 + 9 \times 7$.

2. Calculate $y(2)$ by plugging $x = 2$ into the function: $y(2) = 2^2 + 9 \times 2$.

3. Set up the difference quotient: $v_{av} = \frac{y(7) - y(2)}{7 - 2}$.

4. Perform the subtraction in the numerator and division in the denominator, but stop before calculating the final value.

Try solving on your own before revealing the answer!

Final Answer: $v_{av} = 18$

Plugging in the values: $y(7) = 112$, $y(2) = 22$, so $v_{av} = \frac{112 - 22}{7 - 2} = \frac{90}{5} = 18$.

This is the average velocity over the interval $[2,7]$.

Q2. Find the average velocity of $y = 3x - 2$ over the interval $[5,8]$.

Background

Topic: Average Velocity (Difference Quotient)

This question again tests your ability to compute the average rate of change for a linear function over a specified interval.

Key formula:

$v_{av} = \frac{y(b) - y(a)}{b - a}$

• $y(x)$: The function value at $x$

• $a, b$: The endpoints of the interval

Step-by-Step Guidance

1. Calculate $y(8)$: $y(8) = 3 \times 8 - 2$.

2. Calculate $y(5)$: $y(5) = 3 \times 5 - 2$.

3. Set up the difference quotient: $v_{av} = \frac{y(8) - y(5)}{8 - 5}$.

4. Subtract and divide, but stop before the final calculation.

Try solving on your own before revealing the answer!

Final Answer: $v_{av} = 3$

Plugging in: $y(8) = 22$, $y(5) = 13$, so $v_{av} = \frac{22 - 13}{8 - 5} = \frac{9}{3} = 3$.

This is the average velocity for the interval $[5,8]$.

Q3. Find the average velocity of $y = -3x^2 - x$ over the interval $[3,4]$.

Background

Topic: Average Velocity (Difference Quotient)

This question tests your ability to apply the difference quotient to a quadratic function.

Key formula:

$v_{av} = \frac{y(b) - y(a)}{b - a}$

• $y(x)$: The function value at $x$

• $a, b$: The endpoints of the interval

Step-by-Step Guidance

1. Calculate $y(4)$: $y(4) = -3 \times 4^2 - 4$.

2. Calculate $y(3)$: $y(3) = -3 \times 3^2 - 3$.

3. Set up the difference quotient: $v_{av} = \frac{y(4) - y(3)}{4 - 3}$.

4. Subtract and divide, but stop before the final calculation.

Try solving on your own before revealing the answer!

Final Answer: $v_{av} = -22$

Plugging in: $y(4) = -52$, $y(3) = -30$, so $v_{av} = \frac{-52 - (-30)}{4 - 3} = \frac{-22}{1} = -22$.

This is the average velocity for the interval $[3,4]$.

Q4. Find the average velocity of $y = 2 + \tan t$ over the interval $[-\frac{\pi}{4}, \frac{\pi}{4}]$.

Background

Topic: Average Velocity (Difference Quotient) with Trigonometric Functions

This question tests your ability to apply the difference quotient to a function involving trigonometric expressions.

Key formula:

$v_{av} = \frac{y(b) - y(a)}{b - a}$

• $y(x)$: The function value at $x$

• $a, b$: The endpoints of the interval

Step-by-Step Guidance

1. Calculate $y(\frac{\pi}{4})$: $y(\frac{\pi}{4}) = 2 + \tan(\frac{\pi}{4})$.

2. Calculate $y(-\frac{\pi}{4})$: $y(-\frac{\pi}{4}) = 2 + \tan(-\frac{\pi}{4})$.

3. Set up the difference quotient: $v_{av} = \frac{y(\frac{\pi}{4}) - y(-\frac{\pi}{4})}{\frac{\pi}{4} - (-\frac{\pi}{4})}$.

4. Perform the subtraction and division, but stop before the final calculation.

Try solving on your own before revealing the answer!

Final Answer: $v_{av} = \frac{4}{\pi}$

Plugging in: $y(\frac{\pi}{4}) = 3$, $y(-\frac{\pi}{4}) = 1$, so $v_{av} = \frac{3 - 1}{\frac{\pi}{2}} = \frac{2}{\frac{\pi}{2}} = \frac{4}{\pi}$.

This is the average velocity for the interval $[-\frac{\pi}{4}, \frac{\pi}{4}]$.

Q5. Use the graph of the given function to find $\lim_{x \to 0} f(x)$.

Background

Topic: Limits from Graphs

This question tests your ability to interpret a graph and determine the limit of a function as $x$ approaches a specific value, including checking for the existence of the limit.

Key Terms:

• Right-hand limit: $\lim_{x \to 0^+} f(x)$

• Left-hand limit: $\lim_{x \to 0^-} f(x)$

• Limit exists if both one-sided limits are equal.

Step-by-Step Guidance

1. Examine the graph as $x$ approaches $0$from the right ($x \to 0^+$) and from the left ($x \to 0^-$).

2. Identify the $y$-values the function approaches from each side.

3. Compare the right-hand and left-hand limits to determine if the overall limit exists.

4. If the values are not equal, the limit does not exist at $x = 0$.

Try solving on your own before revealing the answer!

Final Answer: The limit does not exist.

The right-hand limit is $1$, the left-hand limit is$-1$, so$\lim_{x \to 0} f(x)$ does not exist because the one-sided limits are not equal.

Q6. Use the graph of the given function to find $\lim_{x \to 0} f(x)$.

Background

Topic: Limits from Graphs

This question tests your ability to interpret a graph and determine the limit of a function as $x$ approaches a specific value.

Key Terms:

• Right-hand limit: $\lim_{x \to 0^+} f(x)$

• Left-hand limit: $\lim_{x \to 0^-} f(x)$

• Limit exists if both one-sided limits are equal.

Step-by-Step Guidance

1. Examine the graph as $x$ approaches $0$ from both sides.

2. Identify the $y$-values the function approaches from each side.

3. Compare the right-hand and left-hand limits to determine if the overall limit exists.

4. If the values are equal, the limit exists at $x = 0$.

Try solving on your own before revealing the answer!

Final Answer: $\lim_{x \to 0} f(x) = 0$

Both one-sided limits approach $0$, so the limit exists and equals $0$.

Q7. Use the graph of the given function to find $\lim_{x \to 0} f(x)$.

Background

Topic: Limits from Graphs

This question tests your ability to interpret a graph and determine the limit of a function as $x$ approaches a specific value, including checking for the existence of the limit.

Key Terms:

• Right-hand limit: $\lim_{x \to 0^+} f(x)$

• Left-hand limit: $\lim_{x \to 0^-} f(x)$

• Limit exists if both one-sided limits are equal.

Step-by-Step Guidance

1. Examine the graph as $x$ approaches $0$ from the right and left.

2. Identify the $y$-values the function approaches from each side.

3. Compare the right-hand and left-hand limits to determine if the overall limit exists.

4. If the values are not equal, the limit does not exist at $x = 0$.

Try solving on your own before revealing the answer!

Final Answer: The limit does not exist.

The right-hand limit is $1$, the left-hand limit is$-1$, so$\lim_{x \to 0} f(x)$ does not exist because the one-sided limits are not equal.

Q8. Use the graph of the given function to find $\lim_{x \to 0} f(x)$.

Background

Topic: Limits from Graphs

This question tests your ability to interpret a graph and determine the limit of a function as $x$ approaches a specific value.

Key Terms:

• Right-hand limit: $\lim_{x \to 0^+} f(x)$

• Left-hand limit: $\lim_{x \to 0^-} f(x)$

• Limit exists if both one-sided limits are equal.

Step-by-Step Guidance

1. Examine the graph as $x$ approaches $0$ from both sides.

2. Identify the $y$-values the function approaches from each side.

3. Compare the right-hand and left-hand limits to determine if the overall limit exists.

4. If the values are equal, the limit exists at $x = 0$.

Try solving on your own before revealing the answer!

Final Answer: $\lim_{x \to 0} f(x) = -2$

Both one-sided limits approach $-2$, so the limit exists and equals $-2$.

Q9. Use the graph of the given function to find $\lim_{x \to 0} f(x)$.

Background

Topic: Limits from Graphs

This question tests your ability to interpret a graph and determine the limit of a function as $x$ approaches a specific value.

Key Terms:

• Right-hand limit: $\lim_{x \to 0^+} f(x)$

• Left-hand limit: $\lim_{x \to 0^-} f(x)$

• Limit exists if both one-sided limits are equal.

Step-by-Step Guidance

1. Examine the graph as $x$ approaches $0$ from both sides.

2. Identify the $y$-values the function approaches from each side.

3. Compare the right-hand and left-hand limits to determine if the overall limit exists.

4. If the values are equal, the limit exists at $x = 0$.

Try solving on your own before revealing the answer!

Final Answer: $\lim_{x \to 0} f(x) = -2$

Both one-sided limits approach $-2$, so the limit exists and equals $-2$.