5.2 Increasing and decreasing functions questions for IB Maths AI

Write down the $x$-coordinate of the local maximum point of the graph of $f$. Give a reason for your answer.

Write down the $x$-coordinates of the local minimum points of the graph of $f$.

Write down the intervals in which $f$ is decreasing.

Calculate the average rate of change of the volume of water in the reservoir during the first 12 months.

Find ${\displaystyle \frac{dV}{dt}}$.

Find the value of ${\displaystyle \frac{dV}{dt}}$ when $t=6$.

Interpret your answer to part (c)(i) in the context of the question.

Explain why the answer to part (a) gives little information about how the volume of water changed during the first 12 months.

Find the values of $t$ for which the volume of water in the reservoir is decreasing.

The water company issues a drought warning if the volume of water falls below 70 million m³. Determine whether the model predicts a drought warning during these 15 months.

Find $f^{\mathrm{\prime }}(x)$.

Solve $f^{\mathrm{\prime }}(x)=0$.

Hence write down the values of $x$ for which $f$ is decreasing.

Find the concentration of the drug 2 hours after it is taken.

Find the maximum concentration of the drug and the time at which it occurs.

Use your graphic display calculator to find $C^{\mathrm{\prime }}(1)$ and $C^{\mathrm{\prime }}(6)$. Explain what the signs of these values tell you about the concentration of the drug.

Find the average rate of change of the concentration between $t=2.5$ and $t=8$.

Find $C(20)$ and $C(40)$. Hence write down the limit of $C(t)$ as $t\to \mathrm{\infty }$.

A doctor is concerned when the concentration is falling at a rate of more than 1 mg L⁻¹ per hour.

Find the length of time during which the concentration is falling at a rate of more than 1 mg L⁻¹ per hour.

Find $D^{\mathrm{\prime }}(t)$.

Show that, according to the model, the total number of downloads is always increasing.

Interpret the value $D^{\mathrm{\prime }}(2.5)$ in the context of the question.

Write down the value of $r(0)$ and interpret it in the context of the question.

Write down the value of $t$ at which the volume of water has a local minimum. Justify your answer.

At the start of the period, the reservoir holds 50 thousand m³ of water. Let $V(t)$ be the volume of water, in thousands of m³, after $t$ days.

Find an expression for $V(t)$.

Find the greatest volume of water in the reservoir during the 12 days.

Find the least volume of water in the reservoir during the 12 days.

Find the greatest rate at which the volume of water increases during the 12 days.