C1 DIFFERENTIATION Worksheet C 1
Find the gradient at the point with x-coordinate 3 on each of the following curves. a y = x3
b y = 4x − x2
c y = 2x2 − 8x + 3
d y = + 2
3x
2
Find the gradient of each curve at the given point. a y = 3x2 + x − 5 c y = x(2x − 3) e y = x2 + 6x + 8
(1, −1) (2, 2) (−3, −1)
b y = x4 + 2x3 f y = 4x + x−2
(−2, 0) (1, 6) 2d y = x2 − 2x−1 (2, 3)
3 Evaluate f ′(4) when
4
a f(x) = (x + 1)2
b f(x) = x2
1
c f(x) = x − 4x−2
d f(x) = 5 − 6x2
3
The curve with equation y = x3 − 4x2 + 3x crosses the x-axis at the points A, B and C. a Find the coordinates of the points A, B and C.
b Find the gradient of the curve at each of the points A, B and C.
5
For the curve with equation y = 2x2 − 5x + 1, dy
a find ,
dx
b find the value of x for which
dy
= 7. dx
6 7
Find the coordinates of the points on the curve with the equation y = x3 − 8x at which the gradient of the curve is 4.
A curve has the equation y = x3 + x2 − 4x + 1. a Find the gradient of the curve at the point P (−1, 5).
Given that the gradient at the point Q on the curve is the same as the gradient at the point P,
b find, as exact fractions, the coordinates of the point Q. 8
Find an equation of the tangent to each curve at the given point. a y = x2 (2, 4) b y = x2 + 3x + 4 c y = 2x2 − 6x + 8
(1, 4)
d y = x3 − 4x2 + 2
(−1, 2) (3, −7)
9
Find an equation of the tangent to each curve at the given point. Give your answers in the form ax + by + c = 0, where a, b and c are integers. a y = 3 − x2 c y = 2x2 + 5x − 1
(−3, −6) (1, 2) 2
b y = (2, 1) d y = x − 3x (4, −2)
2
x
10
Find an equation of the normal to each curve at the given point. Give your answers in the form ax + by + c = 0, where a, b and c are integers. a y = x2 − 4 c y = x3 − 8x + 4
(1, −3) (2, −4)
b y = 3x2 + 7x + 7
6x
(−2, 5)
d y = x − (3, 1)
11
C1 DIFFERENTIATION Find, in the form y = mx + c, an equation of
Worksheet C continued a the tangent to the curve y = 3x2 − 5x + 2 at the point on the curve with x-coordinate 2, b the normal to the curve y = x3 + 5x2 − 12 at the point on the curve with x-coordinate −3. A curve has the equation y = x3 + 3x2 − 16x + 2.
a Find an equation of the tangent to the curve at the point P (2, −10).
12
The tangent to the curve at the point Q is parallel to the tangent at the point P.
b Find the coordinates of the point Q. 13
A curve has the equation y = x2 − 3x + 4.
a Find an equation of the normal to the curve at the point A (2, 2). The normal to the curve at A intersects the curve again at the point B.
b Find the coordinates of the point B. 14
f(x) ≡ x3 + 4x2 − 18.
a Find f ′(x).
b Show that the tangent to the curve y = f(x) at the point on the curve with x-coordinate −3 passes through the origin.
15 The curve C has the equation y = 6 + x − x2.
a Find the coordinates of the point P, where C crosses the positive x-axis, and the point Q,
where C crosses the y-axis.
b Find an equation of the tangent to C at P.
c Find the coordinates of the point where the tangent to C at P meets the tangent to C at Q. 16
The straight line l is a tangent to the curve y = x2 − 5x + 3 at the point A on the curve. a find the coordinates of the point A,
Given that l is parallel to the line 3x + y = 0,
b find the equation of the line l in the form y = mx + c.
17 18
The line with equation y = 2x + k is a normal to the curve with equation y = Find the value of the constant k.
16x2
.
A ball is thrown vertically downwards from the top of a cliff. The distance, s metres, of the ball from the top of the cliff after t seconds is given by s = 3t + 5t 2.
Find the rate at which the distance the ball has travelled is increasing when a t = 0.6, b s = 54.
19
Water is poured into a vase such that the depth, h cm, of the water in the vase after t seconds is given by h = kt3, where k is a constant. Given that when t = 1, the depth of the water in the vase is increasing at the rate of 3 cm per second,
1
a find the value of k,
b find the rate at which h is increasing when t = 8.
