CAIE A-Level Mathematics AS 2.4.1 Advanced Differentiation Questions

Practise differentiating eˣ, ln x and trigonometric composites with product, quotient and chain rules and solving derivative conditions for exact gradients or stationary points.

Exam points

use vector notation and operations
solve geometric problems with vectors
interpret vector relationships

[Maximum number: 3]

Figure for Question CAIE A-Level Mathematics AS 2.4.1 Advanced Differentiation Questions question 1 — CAIE A-Level Mathematics AS

The diagram shows the curve with equation $y=8 \mathrm{e}^{-x}-\mathrm{e}^{2 x}$ . The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.

Find the gradient of the curve at A.

Differentiate to obtain form $k_{1} \mathrm{e}^{-x}+k_{2} \mathrm{e}^{2 x}$
Where $k_{1} k_{2} \neq 0, k_{1} \neq 8$ and $k_{2} \neq-1$ .

Obtain $-8 \mathrm{e}^{-x}-2 \mathrm{e}^{2 x}$
Substitute x=0 to obtain -10

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CAIE A-Level Mathematics AS 2.4.1 Advanced Differentiation Questions question 1