Algebra Class 10: Exponents and Logarithms

What is the simplest form of the expression $3(x^2 y)y^{-2} z^5$?

When rewriting $ln((z^{-5})^4)$ using properties of logarithms, which expression is correct?

What is the average velocity of the baseball between $t = 0$ and $t = 2$ using the formula $s(t) = -16t^2 + 60t + 4$?

If the instantaneous velocity of the baseball at $t = 1$ is approximated by narrowing intervals, which average velocity interval yields the best estimate?

What is the limit of the function $f(x)$ as $x$ approaches 1?

What can be inferred about the continuity of the piecewise function $f(x)$?

What is the result of simplifying the expression $(81^{1/2})(21^{1/2})$?

What is the expression for $ln(5e^3)$ using properties of logarithms?

Homework Instructions

Complete problems on a separate sheet and upload as PDF to Canvas by September 4 at 10:10 AM.
Bring a physical copy to class on the due date.
Show all work and write legibly.

Question 1: Exponent Simplification

Simplify expressions involving exponents; no negative exponents in the final answer.
Example expressions to simplify include:
- (x²y)y⁻²z⁵ = z³
- (7b⁴)³a⁻²
- (81^(1/2))(2^(1/2))

Question 2: Logarithm Properties

Use properties of logarithms to rewrite expressions as sums, differences, or products of logarithms.
Expressions to rewrite include:
- ln((z⁻⁵)⁴)
- ln(5e³), where e is the irrational number.

Question 3: Motion Model of a Baseball

The position function s(t) = -16t² + 60t + 4 describes the baseball's motion.
Average velocity calculation:
- Find average velocity between t = 0 and t = 2.
- Understand graphical representation of average velocity.
Instantaneous velocity estimate at t = 1:
- Calculate average velocity over three narrowing time intervals around t = 1.
- Clearly state chosen intervals and show calculation steps.
Optional quicker method for finding instantaneous velocity is encouraged.

Question 4: Piecewise-Defined Function Analysis

Evaluate the function f at specific points:
- Find f(1).
- Calculate lim as x approaches 1.
Discuss the existence and value of the limit as x approaches -1; provide reasoning if it does not exist.
Find f(3).
Analyze continuity of function f and identify intervals of continuity.

Test your understanding of exponent simplification and logarithm properties with this quiz. You'll also apply your knowledge of motion models to analyze a baseball's velocity. Each question challenges you to demonstrate your math skills and understanding of these concepts.