1. The first graph below shows the function y ( t ) = sin( t ) and its tangent l

Question

1. The first graph below shows the function y(t) = sin(t) and its tangent line at t = a.

Suppose the slope of the indicated tangent line is m. In math symbols, y'(a) = m

Suppose that you slow the function down to y(t) = sin(0.5t), as shown below.

What's the new tangent slope at the new location, t = 2a? Write your answer in terms of m.

2.

The population of fish in a lake is observed to fluctuate during a six month period according to the formula

P(t) = 80000 + 20000?sin?t

where t is time in months. Answer the questions below.

Note that the units are provided outside the answer boxes. You don't have to include them in your answers.

You may find it useful to graph this function on the domain 0 ? t ? 6 months.

a) How fast is the population changing at the instant t=4 months?

b) What is the smallest number of fish during the six month period?

c) When does the population low point occur? Be accurate to 2 decimal places.

d) Shortly before the low point, the population was 70,000 fish. How fast was the population changing at that instant?

e) Shortly after the low point there is an instant when the population is changing at 10,000 fish/month. When is it? Be accurate to 2 decimal places.

f) If the population is changing at -11,000 fish/month, how many fish are there?
NOTE: There are two possible correct answers. [Larger and smaller number]

Explanation / Answer

1) 0.5*m

2)

a) 19951.28

b) 81395.12

c) 80000

d) 17320

e) 0.5 months

f) 69000

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