7. Solve the differential equation y” + 9y = csc 3x.

A tennis ball bounces on the floor three times, and each time it loses 23.0% of its energy due to heating. How high does it bounce after the third time, if we released it from the floor?

1. Solve (1 + x) dy – y dx = 0 by using separation of variables.

In the figure, block A has a mass of 3.00 kg. It rests on a smooth horizontal table and is connected by a very light horizontal string over an ideal pulley to block B, which has a mass of 2.00 kg. When block B is gently released from rest, how long does it take block B to travel 80.0 cm?

How is generalization used to simplify the models of a system with many similar objects? (Choose all that apply)

Prove, or provide a counterexample to disprove, the following statement:             “The function f : ℕ ⟶ ℕ  be defined by f(n) = n2 + 3 is onto.” Use good proof technique. Grading rubric:1 pt. State the definition of onto at the beginning, then prove or disprove.1 pt. State any givens and assumptions.1 pt. Clearly explain your reasoning.1 pt. Remember to state the final conclusion at the end of the proof. Note:  To avoid the need for typing superscript exponents, you may use the expression ‘n^2’ or ‘n-squared’ to represent n2.

Prove the following statement using induction. “For all positive integers n, 5|(n5 – n).”  Use good proof technique.  Note:  To avoid the need for typing superscript exponents, you may use the expression ‘n^5’ to represent n5. Grading rubric:1 pt. State the basis step, then prove it.1 pt. State the inductive hypothesis.2 pt. Complete the proof of the inductive step.1 pt. State the final conclusion at the end of the proof.1 pt. Label each part: the basis step, inductive hypothesis, inductive step, and conclusion.

Prove that 2×2 + x + 4 is O(x2), by identifying values for C and k and demonstrating that they do satisfy the definition of big-O for this function.  Show your work. Note:  To avoid the need for typing superscript exponents, you may use the notation ‘x^2′ to represent x2.

Define S, a set of integers, recursively as follows: Initial Condition: 0 ∈ SRecursion: If m ∈ S then m + 2 ∈ S. Which of the following sets is equivalent to set S?

The function f : ℝ ⟶ ℝ defined by f(x) = 5×3 + 3×2 – x + 7 is O(x).