CAIE A-Level Mathematics 1.7.3 Applications of Differentiation

Practise using derivatives to construct tangents and normals, test increasing or decreasing behaviour and combine rates for coordinates, arcs, areas and other changing quantities.

Exam points

evaluate dy/dx at the point and use the negative reciprocal for a normal gradient
use the sign of a derivative throughout its domain to classify increasing or decreasing behaviour
apply dQ/dt = (dQ/dx)(dx/dt) and evaluate all quantities at the stated instant

[Maximum number: 4]

Figure for Question 1.7.3—Differentiation applications question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=9\left(x^{-\frac{1}{2}}-4 x^{-\frac{3}{2}}\right)$ . The curve crosses the x-axis at the point A.

Find the equation of the tangent to the curve at A.

$\frac{\mathrm{d} y}{\mathrm{~d} x}=9\left(-\frac{1}{2} x^{-\frac{3}{2}}+6 x^{-\frac{5}{2}}\right)$
B2, 1, 0
B2; all 3 terms correct: $9,-\frac{1}{2} x^{-\frac{3}{2}}$ and $6 x^{-\frac{5}{2}}$
B1; 2 of the 3 terms correct
At x=4 gradient $=9\left(-\frac{1}{16}+\frac{6}{32}\right)=\frac{9}{8}$
M1
Using their x=4 in their differentiated expression
and attempt to find equation of the tangent.
Equation is $y=\frac{9}{8}(x-4)$
A1
or $y=\frac{9 x}{8}-\frac{9}{2} \mathrm{OE}$

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1.7.3—Differentiation applications question 1