This document provides instruction on calculating the gradient of a curve at a point and finding the equation of the tangent line to a curve at a point. It includes examples of finding the gradient at specific points on curves defined by equations and finding the coordinates of points where a curve has a horizontal tangent. It also discusses using calculators or software to find gradients and draw tangents and normals to curves at given points.

1. Gradient at a point
By the end of the lesson you will be able to:
• Calculate the gradient of a curve at any point.
• Find the equation of the tangent line to a curve
at a point.
Tangent line to a curve

2. Find the gradient of the curve
at the point P = ( 1 , 2 ).

3. Find the gradient of the curve
at the point P = ( 2 ,­8 ).
y = x3
­ 4x2

4. Find f '(3) if f(x)= x3
­2x2
+10.

5. The gradient of the curve y = 3 x2
+ x ­ 3 at
the point P is 13. Find the coordinates of P.

6. Given that f(x )= x3
­x2
­x+1, find the coordinates
of all the points for which the curve y=f(x) has a
horizontal tangent.
http://www.ies.co.jp/math/java/calc/doukan/doukan.html

7. Use your GDC to find the gradient of the curve
y = 9 x ­ x2
at the point (2, 14).

8. Find the equation of the tangent to the curve
y= x2
at the point (3,9)
Tangent to a curve

9. at the point P = ( 1 , 2 ).
Find the equation of the tangent to the curve

10. The normal to a curve at a point P is the straight
line through P which is perpendicular to the
tangent at P.
n
t
⇒ gradient of n x gradient of t = ­1

11. If we are traveling in a car around a corner
and we hit something slippery on the road
(like oil, ice, water or loose gravel) and our
car starts to skid, it will continue in a
direction tangent to the curve.
A car has skidded while
rounding a corner,
tangent to the double
yellow lines curve.

12. Find the equation of the normal to the curve
y = 3x2
+7x­2 at the point P where x=­1

13. Use of GDC
Find the equation of the tangent and normal to the
curve at x = 0.5 . Reproduce with GDC.
(Sketch) (tang)
( move the cursor with the right­left arrow until
you reach the point and press to draw the
line
• Draw the function.
• .
• . (Sketch) (normal)
fx9860emulator.exe

14. Book page 375,
Ex 12G: 1a­3­4­6­7­8

15. http://www.univie.ac.at/future.media/moe/galerie/diff1/diff1.html#ableitung
http://www.ies.co.jp/math/java/calc/doukan/doukan.html
http://www.ima.umn.edu/~arnold/calculus/differential/differential­g.html
http://www.ies.co.jp/math/java/calc/difsin2/difsin2.html
http://www.ies.co.jp/math/java/calc/heihen/heihen.html
http://www.math.umn.edu/%7Egarrett/qy/TraceTangent.html
http://www.calculus­help.com/funstuff/phobe.html
Useful links:

16. Attachments
fx9860emulator.exe