Skip to main content

Back to the full dot-point answer

QLDMath MethodsQuick questions

Unit 4: Further calculus and statistical inference

Quick questions on Implicit differentiation and related rates: QCE Maths Methods Unit 4

9short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is the four-step procedure?
Show answer
Step 1. Identify the variables and relate them. Write an equation linking the time-dependent quantities. Often a geometric formula (volume of a sphere, area of a circle, Pythagoras).
What is worked example. Cone tank?
Show answer
A conical tank with height 4 m and top radius 2 m fills with water at 0.5 m3^3/min. Find the rate at which water depth rises when h=1h = 1 m.
What are decreasing rates?
Show answer
If a quantity is decreasing, its rate is negative. "Water draining at 5 L/min" gives dVdt=−5\frac{dV}{dt} = -5. The negative sign carries through the calculation.
What is sphere inflation?
Show answer
V=43πr3V = \frac{4}{3} \pi r^3. dVdt=4πr2drdt\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}.
What is cone water tank?
Show answer
Tank: height HH, top radius RR. Water depth hh, surface radius r=hR/Hr = h R / H. So V=13πr2h=πR23H2h3V = \frac{1}{3} \pi r^2 h = \frac{\pi R^2}{3 H^2} h^3.
What is shadow length from a walker and a fixed light source?
Show answer
Similar-triangle setup; differentiate the linear constraint.
What is chain rule omitted?
Show answer
When differentiating yny^n with respect to xx (with yy a function of xx), the answer is nyn−1dydxn y^{n-1} \frac{dy}{dx}, not nyn−1n y^{n-1}.
What are sign error for decreasing quantities?
Show answer
If volume is decreasing, dVdt\frac{dV}{dt} is negative. Watch the sign.
What is units missing?
Show answer
Related-rates answers must include units, and the units must be consistent throughout the calculation.

All Math MethodsQ&A pages