3

Set 1: Exponential Functions

Explanation

Answer: A

A viral video's views follow V(t)=1,000,0001+999e0.8tV(t) = \frac{1,000,000}{1 + 999e^{-0.8t}} where tt is in days. How many views does it have initially (t=0t=0)?

A.

1,000

✓ Correct
B.

1,000,000

C.

999

D.

500,000

Detailed Explanation

Choice A is correct. Choice A is the correct answer. Evaluate at t=0t = 0. 1. Substitute: V(0)=1,000,0001+999e0V(0) = \frac{1,000,000}{1 + 999 e^{0}}. 2. Simplify: e0=1e^0 = 1, so V(0)=1,000,0001+999=1,000,0001000V(0) = \frac{1,000,000}{1 + 999} = \frac{1,000,000}{1000}. 3. Result: V(0)=1,000V(0) = 1,000 views. Strategic Tip: Logistic functions start at K1+A\frac{K}{1+A} when t=0t=0. Choice B is incorrect because this is the carrying capacity (long-term limit). Choice C is incorrect because this is close but not the exact initial value. Choice D is incorrect because this is half the carrying capacity.

Key Steps:

The correct answer is 1,000

Why others are wrong:
B: Choice B is incorrect and may result from a calculation error.
C: Choice C is incorrect and may result from a calculation error.
D: Choice D is incorrect and may result from a calculation error.

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