Module 2 Mastery Exercises - 1.pdf

2/6/24, 10:07 PM Module 2 Mastery Exercises - Knewton https://www.knewton.com/learn/section/ef980133-667b-4ae2-a540-118a4c17192f/quiz/79317b97-3997-4586-94a7-4aec778e7fc7/results 1/32 Module 2 Mastery Exercises QUESTION 1 · 1/1 POINTS The graph of function is shown below. At which value of is the slope of the tangent line to the curve equal to ? BACK TO OVERVIEW Attempts Attempt 1: 67% (10/15 points), Feb 06 at 9:55pm CST Questions to show: All (15)

2/6/24, 10:07 PM Module 2 Mastery Exercises - Knewton https://www.knewton.com/learn/section/ef980133-667b-4ae2-a540-118a4c17192f/quiz/79317b97-3997-4586-94a7-4aec778e7fc7/results 2/32 That is correct! Answer Explanation Correct answer:

2/6/24, 10:07 PM Module 2 Mastery Exercises - Knewton https://www.knewton.com/learn/section/ef980133-667b-4ae2-a540-118a4c17192f/quiz/79317b97-3997-4586-94a7-4aec778e7fc7/results 3/32 Checking the slope of the tangent at each point, we see that the slope of the tangent at is approximately equal to . So the answer is . QUESTION 2 · 1/1 POINTS Given the function , which of the following is a valid formula for the instantaneous rate of change at ? That is correct! FEEDBACK Content attribution

2/6/24, 10:07 PM Module 2 Mastery Exercises - Knewton https://www.knewton.com/learn/section/ef980133-667b-4ae2-a540-118a4c17192f/quiz/79317b97-3997-4586-94a7-4aec778e7fc7/results 4/32 Answer Explanation Correct answer: Recall that the instantaneous rate of change of a function at is given by Therefore, the instantaneous rate of change of at is given by QUESTION 3 · 1/1 POINTS Find the slope of the secant line between and on the graph of the function . That is correct! FEEDBACK Content attribution −2

2/6/24, 10:07 PM Module 2 Mastery Exercises - Knewton https://www.knewton.com/learn/section/ef980133-667b-4ae2-a540-118a4c17192f/quiz/79317b97-3997-4586-94a7-4aec778e7fc7/results 5/32 Answer Explanation This question is asking for the rate of change, which is the same as the slope between two points on the curve. When , the -value on the curve is So the ±rst point is . When , the -value on the curve is So the second point is . The average rate of change is the slope between these points, The average rate of change between and for the function is . In other words, the secant line that connects these points has slope . QUESTION 4 · 1/1 POINTS Find the values of and that make the following piecewise function di²erentiable everywhere. Correct answers: −2 FEEDBACK Content attribution

2/6/24, 10:07 PM Module 2 Mastery Exercises - Knewton https://www.knewton.com/learn/section/ef980133-667b-4ae2-a540-118a4c17192f/quiz/79317b97-3997-4586-94a7-4aec778e7fc7/results 6/32 That is correct! Answer Explanation Correct answer: and Note that both of the pieces of this piecewise function are polynomials, which are always di²erentiable. The only point we are unsure about is the boundary point . So we need to make sure the function is di²erentiable at . Before we can worry about di²erentiability, we must make sure the function is continuous. The limits from the left and right must be equal for the function to be continuous. The limit of the function from the left is The limit of the function from the right is Setting these limits equal and solving for , we ±nd and and and and and and

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