CAIE IGCSE Additional Math 14.5 Find Gradients Tangents and Normals Questions

Practise using derivatives to calculate gradients, construct tangent and normal equations and solve their intersections with curves, lines and coordinate axes.

Exam points

Use the derivative at a point to find gradients, then form equations of tangents and normals and solve their intersections with curves or axes.

[Maximum number: 9]

The point P lies on the curve y=(5 x+2)^{2/3}.
The x-coordinate of P is 5.
The normal to the curve at P intersects the line x+y=11 at the point Q.
The point R is the reflection of Q in the tangent to the curve at P.
Find the coordinates of R.

[When x=5 ] y=9
$\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2}{3}(5 x+2)^{-\frac{1}{3}} \times 5$ oe
B1 for $\frac{\mathrm{d} y}{\mathrm{~d} x}=k(5 x+2)^{-\frac{1}{3}}, k \neq \frac{10}{3}$ $\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{x=5}=\frac{10}{9}$
FT their $\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{x=5}$ providing previous B1
awarded

Equation of normal e.g.
$y-\text { their } 9=-\frac{1}{\text { their } \frac{10}{9}}(x-5) \text { oe, soi }$
FT $\frac{-1}{\text { their }\left.\frac{\mathrm{d} y}{\mathrm{~d} x}\right|_{x=5}}$ and their y-coordinate

Eliminates one variable e.g.
$11-x-\text { their } 9=-\frac{1}{\text { their } \frac{10}{9}}(x-5)$
dep on previous M mark

For Q: x=-25, y=36
A1 for each

Coordinates of R:(35,-18)
or ((10-their(-25)), (18-their36))
FT their coordinates of Q

All question bank results loaded

CAIE IGCSE Additional Math 14.5 Find Gradients Tangents and Normals Questions question 1