Differential Calculus 201-103-RE
Vincent Carrier
Exercise Sheet 12
3.7 Tangent Line
Find the equation of the tangent line to the curve at the given value of x.
1. y = x3 + 5x2 − 9
x = −4
2. y =
2
(1 − x)3
9
3. y = √
2x − 1
x=5
√
4. y = 4 3 2 − 3x
√
5. y = x x2 + 3
x=1
x+5
6. y = √
x2 + 7
x=2
x = −2
x=3
Find the value(s) of x at which the tangent line is horizontal.
7. y = x3 − 3x2 − 9x
10. y =
x4
x3 − 6
8. y =
x3
3
− 6x2 + 8
x2
11. y = √
x2 − 2
9. y = x3 (4 − 3x)6
12. y =
x2
(x2 + 3)4
3.8 Implicit Differentiation
Find dy/dx.
13. x2 + y 2 = xy
14. x3 + 3xy + y 3 = 1
16. x2 y 3 + x3 y 2 = 5
17.
√
√
19. x y + y x = 7
20.
√
xy = x + y
15. x + y = x2 y 2 + 3
18.
x
= 1 + xy
y
√
21. y x + y = x
y
+ x2 y 2 = 4
x
Find the equation of the tangent line to the curve at the given values of x and y.
22. xy + x2 y 2 = 6
(x, y) = (1, 2)
23.
√
x+y =x−y
(x, y) = (3, 1)
Answers:
1.
dy
= 3x2 + 10x; y = 8x + 39
dx
2.
dy
6
; y = 6x − 14
=
dx
(1 − x)4
3.
dy
1
9
14
; y =− x+
=−
dx
3
3
(2x − 1)3/2
4.
dy
4
; y = −x + 6
=−
dx
(2 − 3x)2/3
5.
dy
5
2x2 + 3
1
=√
; y = x−
dx
2
2
x2 + 3
6.
1
dy
7 − 5x
19
; y =− x+
= 2
dx
8
8
(x + 7)3/2
dy
9x(4 − x)
; x = 0, 4
= 3
dx
(x − 6x + 8)2
7.
dy
= 3(x + 1)(x − 3); x = −1, 3
dx
8.
9.
dy
4 4
= 3x2 (4 − 9x)(4 − 3x)5 ; x = 0, ,
dx
9 3
10.
√
dy
x3 (x3 − 24)
3
; x = 0, 2 3
=
3
2
dx
(x − 6)
12.
6x(x − 1)(x + 1)
dy
=−
; x = −1, 0, 1
dx
(x2 + 3)5
11.
x(x − 2)(x + 2)
dy
=
; x = −2, 2
dx
(x2 − 2)3/2
13.
dy
2x − y
=
dx
x − 2y
14.
dy
x2 + y
=−
dx
x + y2
15.
dy
2xy 2 − 1
=
dx
1 − 2x2 y
16.
y(3x + 2y)
dy
=−
dx
x(2x + 3y)
17.
√
y − 2 xy
dy
= √
dx
2 xy − x
18.
dy
y(1 − y 2 )
=
dx
x(1 + y 2 )
19.
√
√
y(2 x + y)
dy
=− √
√
dx
x( x + 2 y)
20.
dy
y(1 − 2x3 y)
=
dx
x(1 + 2x3 y)
21.
√
dy
2 x+y−y
=
dx
2x + 3y
dy
y
22.
= − ; y = −2x + 4
dx
x
√
dy
2 x+y−1
3
4
23.
= √
; y = x−
dx
2 x+y+1
5
5