First Principles AI

Question 1

A substance decays exponentially following the formula $A(t) = A_0 e^{-0.1 t}$ , where $A_0$ is the initial amount. Find the limit of $A(t)$ as $t \to \infty$ .

Solution Theory

Question 2

A town’s population is modeled by $P(t) = \dfrac{100,000}{1 + 9e^{-0.5 t}}$ . What is the long-term population as $t \to \infty$ .

Solution Theory

Question 3

A company’s weekly sales (in thousands) follow the function $S(t) = \dfrac{500t}{t + 10}$ . What is the limit of $S(t)$ as $t \to \infty$ and what does it represent?

Solution Theory

Question 4

Signal strength $S(d)$ at a distance $d$ from a tower is modeled as $S(d) = \dfrac{1000}{d^2}$ .

(i)

What is the signal strength at $d = 10$ ?

(ii)

What happens to signal strength as $d \to \infty$ ?

Solution Theory

Question 5

A stock price grows according to the function $P(t) = 50e^{0.05 t}$ .

(i)

Find the limit of $P(t)$ as $t \to \infty$ .

(ii)

Calculate the price after 10 years.

Solution Theory

Question 6

The factory’s cost function is given as: $C(x) = \begin{cases} 10x + 500 & \text{if } x \le 100 \\ 12x + 300 & \text{if } x > 100 \end{cases}$ . Is the cost function continuous at $x = 100$ ?

Solution Theory

Question 7

Inflation is modeled by $I(t) = I_0 e^{0.03 t}$ , where $I_0$ is the initial price index and $t$ is the number of years.

(i)

Find the inflation rate after 5 years if $I_0 = 100$ .

(ii)

What is the expected price index after 10 years?

Solution Theory

Question 8

The cost to produce $x$ units is: $C(x) = \begin{cases} 5x + 20 & \text{if } x \le 10 \\ 6x + 10 & \text{if } x > 10 \end{cases}$ . Is the cost function continuous at $x = 10$ ?

Solution Theory

Questions