Spring 2025 MATH 3300 01 Introduction to Math Thought

1. Which of the following sentences are statements?

(b) a^2 + b^2 = c^2

(c) There exist integers a, b, and c such that a^2 = b^2 + c^2

(d) If x^2 = 4, then x =2

(f) For each real number t, sin^2 t + cos^2 t = 1.

(i) 1+tan^2 theta =sec^2 theta

7. Following is a statement of a theorem which can be proven using the quadratic formula. For this theorem, a, b, and c are real numbers. Theorem If f is a quadratic function of the form f(x) ax^2 +bx +c and ac

(a) g (x)= - 8x^2 + 5x - 2

(b) h (x)= - 1/3x^2 + 3x

(c) k (x)= 8x^2 -5x - 7

(e) f (x)= - 4x^2 -3x + 7

(f) F (x) = - x^4 +x^3 + 9

Activities 10. Exploring Propositions. In Progress Check 1.2, we used exploration to show that certain statements were false and to make conjectures that certain statements were true. We can also use exploration to formulate a conjecture that we believe to be true. For example, if we calculate successive powers of 2 (2 ^1 , 2^2 , 2^3 ,2^4 ,2^5 ,....) and examine the units digits of these numbers, we could make the following conjectures among others.

• If n is a natural number, then the units digit of 2^n must be 2, 4, 6, or 8.
• The units digits of the successive powers of 2 repeat according to the pattern "2, 4, 8, 6."

c) Let f (x) = e^2x. Determine the first eight derivatives of this function. What do you observe? Formulate a conjecture that appears to be true. The conjecture should be written as a conditional statement in the form, "If n is a natural number, then..."

4. Construct a know-show table and do a complete proof for each of the following statements:

(b) If m is an odd integer, then 5m +7 is an even integer.

(c) If m and n are odd integers, then mn + 7 is an even integer.

5. Construct a know-show table and do a complete proof for each of the following statements:

(a) If m is an even integer, then 3m^2 + 2m + 3 is an odd integer.

a . 12. (a) See Exercise (11) for the quadratic formula, which gives the solutions to a quadratic equation. Let a, b, and c be real numbers with a 0. The discriminant of the quadratic equation ax^2+bx+c = 0 is defined to be b^ 2- 4ac. Explain how to use this discriminant to determine if the quadratic equation has two real number solutions, one real number solution, or no real number solutions.

(b) Prove that if a, b, and c are real numbers with a > 0 and c

(c) Prove that if a, b, and c are real numbers with a =/0, b > 0, and b