The document discusses how to find the original curve (primitive function) given the derivative (tangent line equation). It states that if the derivative is f'(x)=xn, then the primitive function is f(x)=(x^(n+1))/(n+1)+c. It provides examples such as if f'(x)=3x^4, then f(x)=3x^5/5+c. It also shows how to find the equation of a curve given its gradient function and a point it passes through.

1. The Primitive Function

2. The Primitive Function
If we know the equation of the tangent, how do we find the original
curve?

3. The Primitive Function
If we know the equation of the tangent, how do we find the original
curve?
If f  x   x n , then the primitive function is;
x n1
f x  c
n 1

4. The Primitive Function
If we know the equation of the tangent, how do we find the original
curve?
If f  x   x n , then the primitive function is;
x n1
f x  c
n 1
e.g. i  f  x   3x 4

5. The Primitive Function
If we know the equation of the tangent, how do we find the original
curve?
If f  x   x n , then the primitive function is;
x n1
f x  c
n 1
e.g. i  f  x   3x 4
3x 5
f x  c
5

6. The Primitive Function
If we know the equation of the tangent, how do we find the original
curve?
If f  x   x n , then the primitive function is;
x n1
f x  c
n 1
e.g. i  f  x   3x 4
3x 5
f x  c
5
ii  f  x   6 x3  5x 2  x  2

7. The Primitive Function
If we know the equation of the tangent, how do we find the original
curve?
If f  x   x n , then the primitive function is;
x n1
f x  c
n 1
e.g. i  f  x   3x 4
3x 5
f x  c
5
ii  f  x   6 x3  5x 2  x  2
6 x 4 5x3 x 2
f x     2x  c
4 3 2

8. The Primitive Function
If we know the equation of the tangent, how do we find the original
curve?
If f  x   x n , then the primitive function is;
x n1
f x  c
n 1
e.g. i  f  x   3x 4
3x 5
f x  c
5
ii  f  x   6 x3  5x 2  x  2
6 x 4 5x3 x 2
f x     2x  c
4 3 2
3 4 5 3 1 2
f x  x  x  x  2x  c
2 3 2

9. (iii) Find the equation of the curve which passes through (1,1) and has
a gradient function of 2x + 3

10. (iii) Find the equation of the curve which passes through (1,1) and has
a gradient function of 2x + 3
dy
 2x  3
dx

11. (iii) Find the equation of the curve which passes through (1,1) and has
a gradient function of 2x + 3
dy
 2x  3
dx
y  x 2  3x  c

12. (iii) Find the equation of the curve which passes through (1,1) and has
a gradient function of 2x + 3
dy
 2x  3
dx
y  x 2  3x  c
when x = 1, y = 1

13. (iii) Find the equation of the curve which passes through (1,1) and has
a gradient function of 2x + 3
dy
 2x  3
dx
y  x 2  3x  c
when x = 1, y = 1
i.e. 1  12  3  c
c  3

14. (iii) Find the equation of the curve which passes through (1,1) and has
a gradient function of 2x + 3
dy
 2x  3
dx
y  x 2  3x  c
when x = 1, y = 1
i.e. 1  12  3  c
c  3
 y  x 2  3x  3

15. (iii) Find the equation of the curve which passes through (1,1) and has
a gradient function of 2x + 3
dy
 2x  3
dx
y  x 2  3x  c
when x = 1, y = 1
i.e. 1  12  3  c
c  3
 y  x 2  3x  3
Exercise 10J; 1ace etc, 2bdf, 3aceg, 4bd, 5b, 7ac, 8bdf
9ace, 10b, 12bd, 14a, 15, 17a