Examples and practice
Applications of differentiation
Back to lessonPractice section 1 · Questions 1–6
Classify the turning points of a rational curve · Classify stationary points on two implicit branches · +4 more
Practice section 2 · Questions 7–12
Sketch derivative branches around two vertical asymptotes · Transfer an oblique asymptote to the derivative sketch · +4 more
Practice section 3 · Questions 13–18
Find the area between an exponential tangent and normal · Find an exact triangle area from an implicit normal · +4 more
- Find the area between an exponential tangent and normal
- Find an exact triangle area from an implicit normal
- Connect a spherical balloon’s volume and radius rates
- Find the falling water level in an inverted cone
- Minimise metal for a cylinder with a hemispherical cap
- Maximise an isosceles triangle inscribed in a circle
Practice section 4 · Questions 19–24
Minimise the metal in a fixed-volume cylindrical can · Match a function to its derivative · +4 more
Practice section 5 · Questions 25–30
Recover a function from its gradient and one value · Find horizontal and vertical tangents to a closed curve · +4 more
- Recover a function from its gradient and one value
- Find horizontal and vertical tangents to a closed curve
- Find a logarithmic tangent and a two-intersection slope range
- Use implicit tangents and a normal to bound an area
- Differentiate a triangle constrained to fixed area
- Optimise a rectangle inside a semicircle
Practice section 6 · Questions 31–36
Minimise the cone circumscribed about a sphere · Combine a leaking cone with trapezium optimisation · +4 more
- Minimise the cone circumscribed about a sphere
- Combine a leaking cone with trapezium optimisation
- Sketch a derivative across two vertical asymptotes
- Sketch the gradient across a pole and a minimum
- Differentiate turning points and a stationary inflection
- Read a derivative from an oblique asymptote and two extrema
Practice section 7 · Questions 37–42
Reconstruct a decreasing function with an isolated zero gradient · Determine a rational curve and sketch its derivative exactly · +4 more
- Reconstruct a decreasing function with an isolated zero gradient
- Determine a rational curve and sketch its derivative exactly
- Identify the parabola touched by every line in a family
- Convert units before differentiating a conical pile
- Optimise and differentiate a triangle on an implicit curve
- Optimise a flower-bed under a wall-building time budget
Practice section 8 · Questions 43–44
Maximise a complete globe on a closed cylindrical base · Optimise corner flower crates around a central pond