First Principles AI

Question 1

A car’s position at time $t$ is given by $s(t) = 5t^3 - 3t^2 + t$ . Find the velocity by differentiating the position function with respect to time.

Solution Theory

Question 2

Structural stress on a bridge is modeled by the function $S(x) = 100 - 5x^2$ , where $x$ is the distance from the center of the bridge. Calculate the rate of change of stress at that point.

Solution Theory

Question 3

A company’s revenue function is given by $R(x) = 1000x - 10x^2$ , where $x$ is the number of units produced. The cost function is $C(x) = 300x + 2000$ .

(i)

Find the profit function $P(x)$

(ii)

Determine the marginal profit when $x = 15$

Solution Theory

Question 4

An investment grows according to the function $A(t) = 10000(1 + 0.05t)^3$ , where $A(t)$ is the value of the investment and $t$ is the time in years.

(i)

Find the rate of change of the investment after 8 years.

(ii)

What is the investment value after 8 years?

Solution Theory

Question 5

The position of a particle moving along a line is given by $s(t) = 5t^3 - 12t^2 + 8t$ , where $s(t)$ is the position in meters and $t$ is the time in seconds.

(i)

Determine the velocity of the particle at $t = 4$ seconds

(ii)

Find the acceleration at $t = 4$ seconds

(iii)

When is the particle at rest?

Solution Theory

Question 6

The position of a car traveling along a straight highway is given by $x(t) = 30t^2 - 4t^3$ , where $x(t)$ is the distance traveled in kilometres and $t$ is the time in hours.

(i)

Find the car’s velocity at $t = 3$ hours.

(ii)

Determine the car’s acceleration at $t = 3$ hours

Solution Theory

Question 7

The stress on a beam under a varying load is given by $S(x) = 400x - x^3$ , where $S(x)$ is the stress in pascals (Pa) and $x$ is the distance from the fixed end in metres. Calculate the rate of change of stress at 6 meters.

Solution Theory

Question 8

The cost $C(r)$ to construct a cylindrical tank depends on the radius of the base, and is given by $C(r) = 8000\pi r^2 + \dfrac{150000}{r}$ , where the first term represents the cost of the base and the second term represents the cost of the walls. Determine the rate of change of the cost at $r = 4$ metres.

Solution Theory

Questions