Applications of differentiation · Short practice

Recall

Tangents and normals — objective 1

Maths4U Original · Short practice

The curve is \(y=\left(x - 1\right)^{2} + 22\). Consider the point with \(x=4\).

Find the tangent gradient at \(\displaystyle x=4\).

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

Fresh review variant

Tangents and normals — objective 1

Maths4U Original · Short practice

The curve is \(y=2 \left(x - 2\right)^{2} + 23\). Consider the point with \(x=3\).

Find the tangent gradient at \(\displaystyle x=3\).

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

Method

Connected rates of change — objective 2

Maths4U Original · Short practice

A spherical bubble has radius \(\displaystyle r=1+2t \mathrm{cm}\), where t is measured in seconds. Let \(\displaystyle A=\pi r^{2}\) be its circular cross-sectional area and \(\displaystyle V= \frac{4\pi r^{3}}{3}\) its volume. Evaluate all instantaneous rates at radius 6 cm.

Find \(\displaystyle \frac{\frac{dV}{dt}}{\pi}\) at the same radius.

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

Fresh review variant

Connected rates of change — objective 2

Maths4U Original · Short practice

A spherical bubble has radius \(\displaystyle r=2+3t \mathrm{cm}\), where t is measured in seconds. Let \(\displaystyle A=\pi r^{2}\) be its circular cross-sectional area and \(\displaystyle V= \frac{4\pi r^{3}}{3}\) its volume. Evaluate all instantaneous rates at radius 7 cm.

Find \(\displaystyle \frac{\frac{dV}{dt}}{\pi}\) at the same radius.

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

Apply

Stationary points and second derivatives — objective 3

Maths4U Original · Short practice

Consider \(y=- 36 x + 3 \left(x - 3\right)^{3} + 131\).

Find the height of the local maximum.

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

Fresh review variant

Stationary points and second derivatives — objective 3

Maths4U Original · Short practice

Consider \(y=- 108 x + 4 \left(x - 4\right)^{3} + 456\).

Find the height of the local maximum.

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

Check understanding

Practical maxima and minima — objective 4

Maths4U Original · Short practice

A rectangular growing area lies against a long wall. A total of 32 m of fencing covers the other three sides. The side parallel to the wall has length x and the perpendicular sides each have width y. Treat both dimensions as positive real numbers; ignore gate widths.

Write \(\displaystyle P=32 m\) for the total fencing length.

Find the area when \(\displaystyle y= \frac{P}{5}\).

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.

Fresh review variant

Practical maxima and minima — objective 4

Maths4U Original · Short practice

A rectangular growing area lies against a long wall. A total of 36 m of fencing covers the other three sides. The side parallel to the wall has length x and the perpendicular sides each have width y. Treat both dimensions as positive real numbers; ignore gate widths.

Write \(\displaystyle P=36 m\) for the total fencing length.

Find the area when \(\displaystyle y= \frac{P}{5}\).

Enter one exact number or fraction. These checks support non-calculator practice. Give full reasoning separately for written problems.