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NSWMaths Extension 1Quick questions

Calculus (ME-C1, C2, C3)

Quick questions on Related rates of change: linking changing quantities via implicit differentiation

13short 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 general method?
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1. Draw a diagram and label every variable that changes. 2. Write down an equation that relates the variables.
What is the chain rule in disguise?
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If V=f(r)V = f(r) and rr changes with time, then dVdt=f′(r)drdt\frac{dV}{dt} = f'(r) \frac{dr}{dt}. This is just the chain rule.
What is choosing the relation?
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The trickiest step is finding the right equation linking the variables. Useful approaches:
What is differentiating implicitly?
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Every variable that changes with time gets a d⋅dt\frac{d \cdot}{dt} attached. The product rule, chain rule and quotient rule apply as usual.
What is direction matters?
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A negative rate means the variable is decreasing. State direction explicitly: "the water level is rising at 0.30.3 m/s" or "the shadow is shortening at 1.21.2 m/s".
What is ladder sliding?
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A 1010 m ladder leans against a vertical wall. The base slides away from the wall at 0.50.5 m/s. How fast is the top sliding down when the base is 66 m from the wall?
What is conical tank?
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Water flows into a cone (vertex down) at 44 cm3^3/s. The cone has radius 66 cm at the top and depth 1212 cm. How fast is the water level rising when the depth is 33 cm?
What is shadow problem?
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A 1.81.8 m person walks away from a 66 m streetlight at 1.51.5 m/s. How fast is the tip of the shadow moving?
What is right triangle with one constant side?
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A boat is being pulled toward a dock by a rope wound at 22 m/s. The dock is 44 m above water. How fast is the boat approaching when the rope is 55 m long?
What is substituting instantaneous values too early?
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Differentiate symbolically first. Substituting numerical values for variables that change before differentiating loses their rate dependence.
What is wrong sign for direction?
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If the question says "approaching", the rate is negative if you define distance as positive away. State direction explicitly in the answer.
What is forgetting the constraint?
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In a cone, the radius and height are linked by similar triangles. Substitute this in before differentiating to reduce to one variable.
What is units missing?
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A rate has units like m/s or cm3^3/s. Final answers without units are incomplete. :::

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