Homework 1 Math 140 PDF

Homework 1 Math 140 For this assignment, please complete the following problems on a separate sheet of paper. Upload your work, preferably in PDF format, to Canvas by the due date (Wednesday, Sep. 4 at 10:10 AM). Also, br...

Homework 1 Math 140 For this assignment, please complete the following problems on a separate sheet of paper. Upload your work, preferably in PDF format, to Canvas by the due date (Wednesday, Sep. 4 at 10:10 AM). Also, bring a physical copy of your work to class the day it is due. Be sure to show all your work and to write legibly. Don’t hesitate to ask me questions! Question 1. Write each of the following expressions involving exponents in their simplest form. No negative exponents should appear in your final answer. 3 (𝑥 2 𝑦) 𝑦 −2 𝑧 5 a.) 𝑧3 3𝑎 −2 b.) (7𝑏4) c.) (81/2 )(21/2 ) Question 2. Use the properties of logarithms to rewrite each expression as a sum, difference, or multiple of logarithms. 𝑥 3 𝑦 −2 a.) ln ((𝑧−5)4 ) b.) ln(5ⅇ 3 ) *Note: Here, ⅇ is the irrational number, not a variable. Question 3. The position 𝑠 (in feet) of a baseball 𝑡 seconds after contacting the bat is modeled by the function 𝑠(𝑡) = −16𝑡 2 + 60𝑡 + 4 a.) Determine the average velocity of the ball between the times 𝑡 = 0 and 𝑡 = 2 (show your steps). Graphically, what does your answer represent? b.) Estimate the instantaneous velocity of the ball at time 𝑡 = 1. Use an approach like Example 2 in the Section 2.1 notes. You must find the average velocity of the ball in at least three time intervals that are narrowing around the point 𝑡 = 1. Clearly state the time intervals you choose, the average velocity of the ball on each of those intervals (show work – don’t only write the final answer), and your conclusion regarding the instantaneous velocity at 𝑡 = 1. c.) (Optional) Is there a faster (more convenient) way of getting the answer? If so, demonstrate it and explain your thinking. Homework 1 Math 140 Question 4. Using the graph of a piecewise-defined function 𝑓 shown below, a.) Find 𝑓(1). b.) Find 𝑙𝑖𝑚 𝑓(𝑥). 𝑥→1 c.) Discuss the existence of 𝑙𝑖𝑚 𝑓(𝑥). If it exists, state what the limit is. If the limit does 𝑥→−1 not exist, explain why. d.) Find 𝑓(3). e.) Discuss the continuity of the function 𝑓. On which intervals is the function continuous? What types of discontinuities exist and where? Question 5. Fill in the missing blanks of the following 𝜀 − 𝛿 proof. Goal: Use the 𝜀 − 𝛿 definition of limit to prove that 𝑙𝑖𝑚(2𝑥 − 8) = 2. 𝑥→5 Proof: For each ________, we must show there exists a 𝛿 > 0 such that if 0 < |𝑥 − ___| < 𝛿, then |(2𝑥 − 8) − 2| < 𝜀. Let 𝛿 = ___. Then 0 < |𝑥 − ___| < 𝛿 can be written as 0 < |𝑥 − ___| < ___ This implies that |(2𝑥 − 8) − 2| = |2𝑥 − 10| = 2 ∗ |x − 5| < 2 ∗ __ =𝜀 Homework 1 Math 140 Question 6. Evaluate the following limits. Simplify. a.) 𝑙𝑖𝑚 (𝑥 3 − 4𝑥 2 + 20) 𝑥→−2 3 b.) 𝑙𝑖𝑚 √𝑦 2 + 2 𝑦→5 c.) 𝑙𝑖𝑚𝜋 tan(6𝑧) 𝑧→ 6 𝑥 2 −𝑥−12 d.) 𝑙𝑖𝑚 𝑥→4 𝑥 2 −16 e.) 𝑙𝑖𝑚 ln(𝑦 2 ) 𝑦→ⅇ √𝑧+4−2 f.) 𝑙𝑖𝑚 𝑧→0 𝑧 Question 7. Explain the Squeeze Theorem in your own words (do not simply recite the definition in the notes/textbook).