Math 34A Exam 2 Review – Calculus Concepts and Applications

Q1. Use the log table and 10x graph to find:

• a. $\log(35 + 42)$

• b. $10^{3.65}$

• c. $\log(\sqrt{8562})$

• d. $\log\left(\frac{1}{2.7}\right)$

• e. $\log(100 \times 378)$

• f. $10^{-2.1}$

• g. The solution to $\log(y) = 1.87$

• h. The average rate of change of $10^x$ between $x=0.2$ and $x=0.7$

• i. The solution to $10^x = \frac{1}{59}$

• j. The rate of change of $10^x$ when $x=0.55$

Background

Topic: Logarithms, Exponential Functions, and Rates of Change

These questions test your understanding of logarithmic and exponential functions, their properties, and how to compute values and rates of change using tables and graphs. Some parts also involve interpreting and solving equations involving logs and exponentials.

Key Terms and Formulas

• Logarithm: $\log_b(a)$ is the exponent to which $b$ must be raised to get $a$.

• Exponential Function: $b^x$ is the function where $b$ is the base and $x$ is the exponent.

• Properties:

  • $\log(ab) = \log a + \log b$

  • $\log\left(\frac{a}{b}\right) = \log a - \log b$

  • $\log(a^k) = k \log a$

  • Average Rate of Change: $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$

  • Instantaneous Rate of Change: $f'(x)$ (the derivative)

Step-by-Step Guidance (Example: Part a)

1. First, compute the value inside the logarithm: $35 + 42$.

2. Once you have the sum, use the log table or calculator to find $\log($result$)$.

3. Remember, if you don't have a calculator, you can estimate using the log table provided in your textbook.

Step-by-Step Guidance (Example: Part h)

1. Recall the formula for average rate of change: $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$.

2. Here, $f(x) = 10^x$, $x_1 = 0.2$, $x_2 = 0.7$.

3. Compute $10^{0.7}$ and $10^{0.2}$ using the $10^x$ graph or table.

4. Plug these values into the average rate of change formula.

Try solving on your own before revealing the answer!

Q2. Use logs/antilogs, the log table, and 10x graph to find:

• a. $387 \times 486$

• b. $23 \div 35.3$

Background

Topic: Using Logarithms to Multiply and Divide

These questions test your ability to use logarithms and antilogarithms to perform multiplication and division, especially when calculators are not allowed.

Key Terms and Formulas

• Multiplication: $\log(ab) = \log a + \log b$

• Division: $\log\left(\frac{a}{b}\right) = \log a - \log b$

• Antilogarithm: If $\log x = y$, then $x = 10^y$

Step-by-Step Guidance (Example: Part a)

1. Find $\log(387)$ and $\log(486)$ using the log table.

2. Add the two logarithms together: $\log(387) + \log(486)$.

3. Find the antilog (use the $10^x$ graph or table) to get the product.

Try solving on your own before revealing the answer!

Q3. Solve the following equations. Leave logs in your answer.

• a. $7^{4x+1} - 3 = 2$

• b. $5^{3x+1} + 2 = 9$

• c. $3 \times 6^{5x-4} = 8$

• d. $2^x = 5(7)^x$

Background

Topic: Solving Exponential Equations Using Logarithms

These problems require you to isolate the exponential term and use logarithms to solve for the variable. You may need to use properties of exponents and logs.

Key Terms and Formulas

• To solve $a^{bx+c} = d$, take $\log$ of both sides.

• Use properties: $\log(a^k) = k \log a$

Step-by-Step Guidance (Example: Part a)

1. Add 3 to both sides to isolate the exponential: $7^{4x+1} = 5$.

2. Take $\log$ of both sides: $\log(7^{4x+1}) = \log(5)$.

3. Use the property $\log(a^k) = k \log a$ to bring the exponent down.

4. Solve for $x$ in terms of logs, but do not simplify further.

Try solving on your own before revealing the answer!

Q4. Solve the equation and simplify your answer without logs: $e^{2x} = e^{10}$

Background

Topic: Solving Exponential Equations with Base $e$

This question tests your ability to solve equations involving the natural exponential function and to simplify the solution.

Key Terms and Formulas

• If $e^a = e^b$, then $a = b$.

Step-by-Step Guidance

1. Recognize that both sides have the same base ($e$), so set the exponents equal: $2x = 10$.

2. Solve for $x$ by dividing both sides by 2.

Try solving on your own before revealing the answer!

Q5. Compute each logarithm:

• a. $\log_4(64)$

• b. $\log_{25}(5)$

• c. $\log_2\left(\frac{1}{16}\right)$

• d. $\ln(e^2)$

Background

Topic: Evaluating Logarithms

These problems test your understanding of logarithm properties and how to compute logarithms with different bases.

Key Terms and Formulas

• Change of base: $\log_b(a) = \frac{\ln a}{\ln b}$

• $\log_b(b^k) = k$

• $\ln(e^k) = k$

Step-by-Step Guidance (Example: Part a)

1. Express 64 as a power of 4 if possible.

2. Use the property $\log_b(b^k) = k$ to evaluate.

Try solving on your own before revealing the answer!

Q6. A bank pays 9% interest compounded annually. There are $1,000 in the account in January 2017. In what year will there first be $10,000 in the account?

Background

Topic: Compound Interest and Exponential Growth

This question tests your ability to use the compound interest formula to solve for time.

Key Terms and Formulas

• Compound Interest: $A = P(1 + r)^t$

• $A$ = final amount, $P$ = principal, $r$ = interest rate per period, $t$ = number of periods

Step-by-Step Guidance

1. Set up the equation: $10,000 = 1,000(1 + 0.09)^t$.

2. Divide both sides by 1,000 to isolate the exponential term.

3. Take the logarithm of both sides to solve for $t$.

4. Use the property $\log(a^t) = t \log a$ to bring down the exponent.

Try solving on your own before revealing the answer!

Q7. A bird population on an island doubles every 6 months. The initial size of the population is 450 birds.

• a. How many birds will there be in 18 years? (No need to simplify your answer.)

• b. How many years would it take the population to increase to 1900? (Leave logs in your answer.)

Background

Topic: Exponential Growth and Doubling Time

These questions test your ability to model population growth using exponential functions and solve for time or population size.

Key Terms and Formulas

• Doubling formula: $P(t) = P_0 \times 2^{t/T}$, where $T$ is the doubling period.

Step-by-Step Guidance (Example: Part a)

1. Calculate the number of doubling periods in 18 years (convert years to months if needed).

2. Plug the values into the formula: $P(18) = 450 \times 2^{\text{number of doubling periods}}$.

Try solving on your own before revealing the answer!

Q8. The half-life of a certain element is 700 years. How long will it take to have 33% of the original amount remaining? (Leave any exponents or logs in your answer)

Background

Topic: Exponential Decay and Half-Life

This question tests your ability to use the half-life formula to solve for time when a certain percentage remains.

Key Terms and Formulas

• Decay formula: $N(t) = N_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}$

• $N(t)$ = amount remaining, $N_0$ = initial amount, $T_{1/2}$ = half-life

Step-by-Step Guidance

1. Set up the equation: $0.33 N_0 = N_0 \left(\frac{1}{2}\right)^{t/700}$.

2. Divide both sides by $N_0$ to simplify.

3. Take the logarithm of both sides to solve for $t$.

4. Use properties of logs to isolate $t$.

Try solving on your own before revealing the answer!

Q9. The number of people who will develop an infectious disease depends on the percentage of people inoculated against it. If $x$ is the percentage of people inoculated, then $f(x)$ people get the disease.

• a. What does $f(0) = 106$ mean?

• b. What does $f'(50) = -103$ mean?

Background

Topic: Function Interpretation and Derivatives

These questions test your ability to interpret the meaning of a function value and its derivative in context.

Key Terms and Formulas

• $f(x)$: Number of people who get the disease when $x$% are inoculated.

• $f'(x)$: Rate of change of the number of people getting the disease with respect to $x$.

Step-by-Step Guidance (Example: Part b)

1. Recall that $f'(50)$ represents the instantaneous rate of change at $x=50$.

2. Interpret the negative value in the context of the problem (what does a decrease mean?).

Try solving on your own before revealing the answer!

Q10. If a TV is priced at $p$ per set, then the number of TVs sold each week is $Q(p)$. What does $Q'(50) = -201$ mean?

Background

Topic: Derivative as Rate of Change

This question tests your ability to interpret the meaning of a derivative in a real-world context.

Key Terms and Formulas

• $Q(p)$: Number of TVs sold per week at price $p$.

• $Q'(p)$: Rate of change of TVs sold with respect to price.

Step-by-Step Guidance

1. Recall that $Q'(50)$ is the derivative at $p=50$.

2. Interpret the negative value in terms of sales and price.

Try solving on your own before revealing the answer!

Q11. Find the average rate of change for $f(x) = 2x^3 - x + 1$ between the following:

• a. $x=1$ and $x=2$

• b. $x=1$ and $x=1.5$

• c. $x=1$ and $x=1.1$

• d. $x=1$ and $x=1.01$

• e. Make an educated guess as to the value of $f'(1)$

Background

Topic: Average Rate of Change and Estimating Derivatives

These questions test your ability to compute average rates of change and use them to estimate the derivative at a point.

Key Terms and Formulas

• Average Rate of Change: $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• Derivative: $f'(x)$ is the instantaneous rate of change at $x$.

Step-by-Step Guidance (Example: Part a)

1. Compute $f(2)$ and $f(1)$ using the given function.

2. Subtract $f(1)$ from $f(2)$.

3. Divide by $2 - 1$ to get the average rate of change.

Try solving on your own before revealing the answer!

Q12. A ferret runs along the x-axis. It is $f(t) = t^2 + 3$ cm from the origin after $t$ seconds. Find the instantaneous velocity after 4 seconds by calculating the average velocity over the interval from $t=4$ to $t=4+\Delta t$ and then taking the limit as $\Delta t \to 0$.

Background

Topic: Instantaneous Rate of Change (Derivative Definition)

This question tests your understanding of the definition of the derivative as the limit of average velocity (difference quotient).

Key Terms and Formulas

• Average velocity: $\frac{f(4 + \Delta t) - f(4)}{\Delta t}$

• Instantaneous velocity: $\lim_{\Delta t \to 0} \frac{f(4 + \Delta t) - f(4)}{\Delta t}$

Step-by-Step Guidance

1. Write the expression for average velocity: $\frac{f(4 + \Delta t) - f(4)}{\Delta t}$.

2. Substitute $f(t) = t^2 + 3$ into the expression.

3. Simplify the numerator: $(4 + \Delta t)^2 + 3 - [4^2 + 3]$.

4. Expand and simplify the numerator as much as possible.

5. Set up the limit as $\Delta t \to 0$.

Try solving on your own before revealing the answer!