Math 34A Exam 2 Review – Calculus Concepts and Problem-Solving Guidance

Study Guide - Smart Notes

Tailored notes based on your materials, expanded with key definitions, examples, and context.

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

a.
b.
c.
d.
e.
f.
g. The solution to
h. The average rate of change of between and
i. The solution to
j. The rate of change of when

Background

Topic: Exponential and Logarithmic Functions, Average and Instantaneous Rate of Change

These questions test your understanding of logarithms, exponentials, and how to use them to solve equations. Parts (h) and (j) also introduce the calculus concepts of average and instantaneous rate of change (derivatives).

Key Terms and Formulas

Logarithm: is the exponent to which must be raised to get .
Exponential: means multiplied by itself times.
Average Rate of Change:
Instantaneous Rate of Change (Derivative):
Properties of Logs: , ,

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

Identify the function: .
Recall the formula for average rate of change between and :
Plug in and :
Evaluate and using a calculator or log table, but do not compute the final value yet.

Try solving on your own before revealing the answer!

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

Background

Topic: Logarithmic Properties and Calculations

These questions test your ability to use logarithms to multiply and divide numbers, and to use antilogs (exponentials) to find the result.

Key Terms and Formulas

Product Rule:
Quotient Rule:
Antilog: If , then

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

Write as .
Use the product rule: .
Find and using the log table.
Add the two logarithms together to get , then use the antilog (exponential) to find the product: .

Try solving on your own before revealing the answer!

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

Background

Topic: Solving Exponential Equations Using Logarithms

These problems require you to isolate the exponential term and use logarithms to solve for the variable.

Key Terms and Formulas

Take logs of both sides to bring exponents down:
Isolate the variable using algebraic manipulation

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

Add $3
Take the logarithm of both sides:
Use the power rule:
Solve for by isolating it, but do not simplify to a final value yet.

Try solving on your own before revealing the answer!

Q4. Solve the equation and simplify your answer without logs:

Background

Topic: Solving Exponential Equations with Base

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

Key Terms and Formulas

If , then

Step-by-Step Guidance

Recognize that both sides have the same base (), so set the exponents equal:
Solve for by dividing both sides by $2$.

Try solving on your own before revealing the answer!

Q5. Compute each logarithm:

Background

Topic: Evaluating Logarithms

These questions test your understanding of the definition of logarithms and their properties.

Key Terms and Formulas

Change of base:
Logarithm properties:
Natural log:

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

Rewrite $64 if possible.
Use the property 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: Exponential Growth and Compound Interest

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

Key Terms and Formulas

Compound Interest:
= final amount, = principal, = interest rate per period, = number of periods

Step-by-Step Guidance

Set up the equation:
Divide both sides by to isolate the exponential term.
Take the logarithm of both sides to solve for .
Use the properties of logarithms to bring down the exponent and solve for .

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

This question tests your ability to model population growth using exponential functions and solve for time using logarithms.

Key Terms and Formulas

Doubling formula: , where is the doubling period
To solve for , use logarithms

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

Set up the equation: (since 6 months = 0.5 years)
Divide both sides by $450$ to isolate the exponential term.
Take the logarithm of both sides to solve for .
Use properties of logarithms to bring down the exponent and solve for (leave logs in your answer).

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, using logarithms.

Key Terms and Formulas

Half-life formula: , where is the half-life
To solve for , use logarithms

Step-by-Step Guidance

Set up the equation:
Divide both sides by to isolate the exponential term.
Take the logarithm of both sides to solve for .
Use properties of logarithms to bring down the exponent and solve for (leave logs in your answer).

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 is the percentage of people inoculated, then people get the disease.

a. What does mean?
b. What does mean?

Background

Topic: Function Interpretation and Derivatives

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

Key Terms and Formulas

: Number of people who get the disease when % are inoculated
: Rate of change of the number of people who get the disease with respect to

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

Recall that represents the rate of change of when .
Interpret the negative value: a negative derivative means the number of people getting the disease decreases as more people are inoculated.
Relate the value to the context: for each 1% increase in inoculation at , the number of people who get the disease decreases by .

Try explaining in your own words before revealing the answer!

Q10. If a TV is priced at per set, then the number of TVs sold each week is . What does mean?

Background

Topic: Interpretation of Derivatives in Context

This question tests your ability to interpret the meaning of a derivative in a business context.

Key Terms and Formulas

: Number of TVs sold per week at price
: Rate of change of TVs sold with respect to price

Step-by-Step Guidance

Recall that represents the rate of change in TVs sold when the price is $50$.
Interpret the negative value: as price increases, the number of TVs sold decreases.
Relate the value to the context: for each p=50 fewer TVs are sold per week.

Try explaining in your own words before revealing the answer!

Q11. Find the average rate of change for the function between the following:

a. and
b. and
c. and
d. and
e. Make an educated guess as to the value of

Background

Topic: Average Rate of Change and Estimating Derivatives

This question tests 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:
Derivative: is the instantaneous rate of change at

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

Compute and using the given function.
Plug these values into the average rate of change formula:
Simplify the numerator and denominator, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q12. A ferret runs along the x-axis. It is cm from the origin after seconds. Find the instantaneous velocity of the ferret after 4 seconds by calculating the average velocity over the interval from to seconds and then taking the limit as .

Background

Topic: Instantaneous Rate of Change (Derivative via Difference Quotient)

This question tests your understanding of how to compute the derivative from first principles using the difference quotient.

Key Terms and Formulas

Difference Quotient:
Instantaneous velocity:

Step-by-Step Guidance

Write the difference quotient for at :
Compute and
Subtract from and simplify the numerator.
Divide by and simplify, but do not take the limit yet.