Flashcards · 8 cards

Applications of differentiation

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Know it
Where is $f(x)$ increasing, and where decreasing?
Answer
Increasing where $f'(x) > 0$ ; decreasing where $f'(x) < 0$ .
From the lesson: Stationary points and optimisation
Rule
How do you find and classify a turning point?
Answer
Solve $f'(x) = 0$ . A maximum if $f''(x) < 0$ (the gradient goes + 0 −); a minimum if $f''(x) > 0$ (− 0 +).
Maximum and minimumFlat tangents: the gradient goes + 0 − at a maximum and − 0 + at a minimum
From the lesson: Stationary points and optimisation
Rule
What goes into a sketch of a cubic?
Answer
The intercepts ( $x = 0$ and $y = 0$ ), the turning points ( $\dfrac{dy}{dx} = 0$ ) and where the ends go, each point labelled. No table of values.
Sketching a cubicPut x = 0 and y = 0 for the intercepts, find the turning points, then show where the ends go
From the lesson: Stationary points and optimisation
Rule
The steps of a best-value (optimisation) problem?
Answer
Write the quantity in one letter, using the given fact to remove the other. Differentiate, set to zero, check it is a max or min, then answer what was asked (the area or volume, not just $x$ ).
1. write the quantity in one letter, using the given fact2. differentiate and set it = 03. solve, then check it is a max or min4. answer the question asked
A best-value problemOne letter, differentiate, set to zero, check, answer
From the lesson: Stationary points and optimisation
Rule
Connected rates of change?
Answer
$\dfrac{dV}{dt} = \dfrac{dV}{dr} \times \dfrac{dr}{dt}$ : the derivative from the formula, times the rate you are given.
dA/dt = dA/dr × dr/dt
from the formula × the rate you are given
Connected ratesThe derivative from the formula, times the rate you are given
From the lesson: Tangents, rates of change and small changes
Rule
Small changes: $y = kx^n$ and $x$ increases by $p\%$ . What happens to $y$ ?
Answer
$y$ increases by about $np\%$ . In general, $\delta y \approx \dfrac{dy}{dx}\,\delta x$ .
δy ≈ (dy/dx) × δx
y = kxⁿ: % change in y ≈ n × % change in x
for small changes the tangent is a good copy of the curve
Small changesy = kxⁿ: the percentage change is about n times as big
From the lesson: Tangents, rates of change and small changes
Which method?
WAEC 2022 · Paper 2 · Q10
A solid rectangular block has a base which measures $3x\text{ cm}$ by $2x\text{ cm}$ . The height of the block is $y\text{ cm}$ and its volume is $72\text{ cm}^3$ .
Express $y$ in terms of $x$ .
Find an expression for the total surface area of the block in terms of $x$ only;
Find the value of $x$ for which the total surface area has a stationary value (2 d.p.).
How do you write the surface area in one letter?
Answer
Use the volume: $6x^2y = 72$ gives $y = \dfrac{12}{x^2}$ . Put that into the surface area, then differentiate.
From the lesson: Stationary points and optimisation Try the question
Which method?
WAEC 2013 · Paper 2 · Q3
A side of a rectangle is three times the other. If the perimeter increases by $2\%$ , find the percentage increase in the area of the rectangle.
How is the change in perimeter linked to the change in area?
Answer
The perimeter $8x$ is a fixed multiple of $x$ , so $x$ rises by 2%. The area $3x^2$ has power 2, so it rises by about $2 \times 2\% = 4\%$ .
From the lesson: Tangents, rates of change and small changes Try the question