INTRODUCTION TO CALCULUS
1)
Sketch the tangent (gradient) function for the following function:
y
x
x1
2)
x2
Sketch the derivative function of the function below.
y
x
x1
x2
x3
3)
y
A
4)
B
C
D
E
F
G
x
At what points on the above graph is the function not differentiable?
Sketch a function that could have the following as its gradient function:
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y
x
x1
5)
6)
7)
8)
9)
10)
11)
12)
13)
x2
Differentiate x2 - 3x + 1 from first principles.
Find the derivative of 5x4 + 3x3 - 2x2 + 7x - 4.
Differentiate x5 + 2x2 - 8x + 9.
Find f 1 (x) if f(x) = 3x11 - x7 + 5x6 + 2x3 + 1.
dy
If y = 8x3 - 2x + 5, find
when x = 2.
dx
dh
If h = 2t3 + t - 3, find
.
dt
dV
Find
if V = 4  r2.
dr
dh
If h = 5t2 + 3t - 7, find
when t = 3.
dt
Differentiate x-5.
2
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
Find the derivative of x 3 .
1
Differentiate 4 .
3x
Differentiate (3x - 7)6.
Find f 1 (x) when f(x) = (x2 + 5x - 2)3.
Differentiate (5x + 1)(3x - 2) by
(a) expanding
(b) using the product rule.
Differentiate 3x2.(7x - 1)5.
4x  3
Differentiate
.
3x  1
x2  5
Differentiate
.
2x  5
Find the gradient of the tangent to the curve y = 2x2 - 3x + 6 at the point (-1,11).
Find the gradient of the normal to the curve y = x3 + 4x2 - x at the point where
x = 1.
Find the equation of the tangent to the curve y = x3 at the point (2,8).
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25)
26)
27)
28)
29)
Find the equation of the tangent to the curve y = 2x2 - 3x + 5 at the point
where x = -3.
Find the equation of the normal to the curve y = x2 + 2 at the point (3,11).
Find the equation of the normal to the curve y = 4x3 - 7x2 + 3 at the point
where x = 2.
Find the gradient of the (a) tangent and (b) normal to the curve y = x2 at
the point (-4,16).
The equation of the normal to a curve at point P is given by 3x - 2y + 5 = 0.
(a) Find the gradient of the tangent to the curve at point P.
dy
(b) If
 2 x  1 , find the coordinates of P.
dx
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ANSWERS
1)
y
x
x2
x1
2)
y
x
x2
x1
3)
4)
x3
D,E,F
y
x1
5)
x2
x
2x - 3
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6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
20x3 + 9x2 - 4x + 7
5x4 + 4x- 8
33x10 - 7x6 + 30x5 + 6x2
94
6t2 + 1
8 r
33
-5x-6
1
2 3
x
3
4
 5
3x
18(3x - 7)5
3(2x + 5)(x2 + 5x - 2)2
30x - 7
105x2(7x - 1)4 + 6x(7x - 1)5 = 3x(7x - 1)4(49x - 2)
13
( 3x  1) 2
2 x 2  10x  10
( 2 x  5) 2
-7
1

10
12x - y - 16 = 0
15x + y + 13 = 0
x + 6y - 69 = 0
x + 20y - 142 = 0
1
(a) -8 (b)
8
2
 5 1
(a) 
(b) P =   ,1 
 6 4
3
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