Michael is very fascinated with how otters play in the water. Michael is watching an Otter named Quad, simply

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1. (15 pts) Michael is very fascinated with how otters play in the water. Michael is watching an Otter named Quad, simply because one day Quad was on this huge rock that was 10 𝑓𝑡 above Quad’s favorite swimming hole, and dove down into the swimming hole and came back up all in a parabolic manner, with a fish in her mouth. Quad’s distance from the surface of the water (in 𝑓𝑡) after 𝑠 seconds from her dive can be modeled by the following function:

𝑄(𝑠) = 10 𝑠2 − 20𝑠 + 10

a. Find 𝑄(1) and interpret it’s meaning in context of the story using a full sentence. (no calculus here, but Show ALL Work.)

b. It turns out that a fish was traveling tangent to Quad’s 𝑄(1) position. Find the tangent line that the fish was following. (Show ALL Work)

c. What was Quad’s maximum depth during this event? (use Calculus, Show ALL Work)

d. Sketch a picture of the scenario (Don’t draw a graph! I want to see some creativity here.)

e. Turns out that Michael is not great at naming otters, what would you have named the otter?

2. (10 points) True/False with Justification! Please decide whether the following statements are true or false and justify your decision with mathematical reasoning.

a. Every function has both an absolute maximum and absolute minimum on any closed interval included in its domain.

b. The only critical values a function has are all the 𝑥-values such that 𝑓′(𝑥) = 0.

c. If a given function is continuous and differentiable on an interval, The Mean Value Theorem implies that there is at least one point on the interval where the tangent line has the same slope as the average rate of change connecting the endpoints of the interval.

d. If 𝑐 is a critical number of 𝑓 and 𝑓′′(𝑐) > 0, then 𝑓 has a local maximum value at 𝑥 = 𝑐.

e. One example of a Strong Form is 00.

3. (10 points) Find 𝑑𝑦/𝑑𝑥 for the following: (Show ALL Work)

a. cos(𝑦) + 𝑥3 + 𝑥𝑦2 = 42

b. ln(𝑦) − 5𝑥−2 + 3𝑦 = −2

4. (5 pts) My ZigZag sewing pattern. The other day I was sewing two pieces of fabric together using a zigzag sewing pattern. It turns out the needle’s position (in millimeters

𝑚𝑚) on the fabric after 𝑡 milliseconds (𝑚𝑠) had passed can be modeled by the following function

𝑁(𝑡) = sin(8𝑡) , 0 ≤ 𝑡

a. Find the velocity function 𝑣(𝑡). (Show ALL Work)

b. Find 𝑣(1) (Show ALL Work) and interpret it’s meaning in a full sentence.

c. Find the acceleration function 𝑎(𝑡). (Show ALL Work)

d. Find 𝑎(1) (Show ALL Work) and interpret it’s meaning in a full sentence.

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