Differentiation

WMA14Topic 4.53 questions

Parametric and implicit differentiation, and forming differential equations from rates of change.

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Questions

Multiple choice: 1
Calculation: 2
Total: 3

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Sample questions

Original practice questions written in the style of the exam.

Question 13 marks
A spherical balloon is being inflated so that its volume increases at a constant rate of $50\,\text{cm}^3\,\text{s}^{-1}$ .

Find the rate at which the radius is increasing when the radius is $5\,\text{cm}$ . Give your answer to 3 significant figures.

(Volume of a sphere $V = \tfrac{4}{3}\pi r^3$ .)
Mark scheme
- B1 $\dfrac{dV}{dr} = 4\pi r^2$ (1)
- M1 Uses (1)
Question 26 marks
The curve $C$ has equation $x^2 + 3xy + y^2 = 11$ .

\frac{d r}{d t}

=

\frac{d V}{d t}

\div

\frac{d V}{d r}

=

\frac{50}{4 π (5^{2})}

\dfrac{dr}{dt} = \dfrac{dV}{dt} \div \dfrac{dV}{dr} = \dfrac{50}{4\pi(5^2)}

\frac{d r}{d t} = \frac{d V}{d t} \div \frac{d V}{d r} = \frac{50}{4 π ( 5 ^{2} )}

\dfrac{dr}{dt} = \dfrac{1}{2\pi} = 0.159\,\text{cm s}^{-1}

(1)

Find an equation of the tangent to $C$ at the point $(1, 2)$ , giving your answer in the form $ax + by + c = 0$ , where $a$ , $b$ and $c$ are integers.

Mark scheme

M1 Differentiates $y^2$ implicitly to $2y\dfrac{dy}{dx}$ (1)
M1 Uses the product rule on $3xy$ : $3y + 3x\dfrac{dy}{dx}$ (1)
A1 $2x + 3y + 3x\dfrac{dy}{dx} + 2y\dfrac{dy}{dx} = 0$ (1)
M1 Substitutes $(1, 2)$ and solves for $\dfrac{dy}{dx}$ (1)
A1 $\dfrac{dy}{dx} = -\tfrac{8}{7}$ (1)
A1 $8x + 7y - 22 = 0$ (1)

dxdy