Applications of Differentiation
∎ Stationary point & Nature:
1. Find the coordinates of the stationary points on each of the following curves and determine the nature of
each of the stationary points.
(a) 𝑦 = 𝑥 2 − 12𝑥 + 8
(b) 𝑦 = (5 + 𝑥)(1 − 𝑥)
(c) 𝑦 = 𝑥 3 − 12𝑥 + 2
(d) 𝑦 = 𝑥 3 + 𝑥 2 − 16𝑥 − 16
(e) 𝑦 = 𝑥(3 − 4𝑥 − 𝑥 2 )
2. Find the coordinates of the stationary points on each of the following curves and determine the nature of
each of the stationary points.
(a) 𝑦 = √𝑥 +
4
√𝑥
2
4
(b) 𝑦 = 𝑥 2 − 𝑥
(c) 𝑦 = 𝑥 + √𝑥
𝑥2
2𝑥
(d) 𝑦 = 𝑥 2 +9
(e) 𝑦 = 𝑥+1
𝑏
3. The curve = 𝑎𝑥 + 𝑥 2 , where 𝑎 and 𝑏 are integers, has a stationary point at (2, 3). Find the value of 𝑎 and
of 𝑏 in this case and determine the nature of the stationary point.
𝑎
𝑥
4. The curve 𝑦 = 𝑥 2 + + 𝑏 has a stationary point at (1, −1).
(a) Find the value of 𝑎 and the value of 𝑏.
(b) Determine the nature of the stationary point (1, −1).
𝑏
5. The curve 𝑦 = 𝑎𝑥 + 𝑥 2 has a stationary point at (−1, −12).
(a) Find the value of 𝑎 and the value of 𝑏.
(b) Determine the nature of the stationary point (−1, −12).
6. A curve is such that =
𝑑𝑦
(a) Show that 𝑑𝑥 = −
𝐴𝑥 2 +𝐵
𝑥 2 −2
2𝑥(2𝐴+𝐵)
(𝑥 2 −2)2
, where 𝐴 and 𝐵 are constants.
.
𝑑𝑦
(b) It is given that 𝑦 = −3 and 𝑑𝑥 = −10 when 𝑥 = 1. Find the values of 𝐴 and of 𝐵.
(c) Using your values of 𝐴 and 𝐵, find the coordinates of the stationary point on the curve, and determine
the nature of this stationary point
𝑑𝑦
7. (a) Given that 𝑦 = (𝑥 2 − 1)√5𝑥 + 2, show that 𝑑𝑥 =
𝐴𝑥 2 +𝐵𝑥+𝐶
,
2√5𝑥+2
where 𝐴, 𝐵 and 𝐶 are integers.
(b) Find the coordinates of the stationary point of the curve 𝑦 = (𝑥 2 − 1)√5𝑥 + 2, for 𝑥 > 0.
Give each coordinate correct to 2 significant figures.
FAISAL MIZAN
1
(c) Determine the nature of this stationary point.
8. A curve has equation 𝑦 = 32𝑥 2 +
1
8𝑥 2
[May20/V12/Q7]
where 𝑥 ≠ 0.
(a) Find the coordinates of the stationary points of the curve.
(b) These stationary points have the same nature. Use the second derivative test to determine whether
they are maximum or minimum points.
[May23/V22/Q2]
Answers:
2
22
[1](a) (6, −28) min, (b) (−2, 9) max, (c) (−2, 18) max, (2, −14) min, (d) (−2 3 , 14 27) max, (2, −36)
1 14
1
min, (e) (−3, −18) min, (3 , 27) max, [2] (a) (4, 4) min, (b) (−1, 3) min, (c) (4, 3) min, (d) (−3, − 3) min,
1
(3, 3) max, (e) (−2, −4) max, (0, 0) min, [3] 𝑎 = 1, 𝑏 = 4; min at 𝑥 = 2, [4] (a) 𝑎 = 2, 𝑏 = −4, (b) min,
1
2
[5] (a) 𝑎 = 8, 𝑏 = −4, (b) max, [6] (b) 𝐴 = 2, 𝐵 = 1, (c) (0, − ) ; max, [7] (a)
25𝑥 2 +8𝑥−5
, (b) (0.315, −1.70),
2√5𝑥+2
(c) min, [8](a) (0.25, 4), (−0.25, 4), (b) minimum.
∎ Gradient:
1. Find the gradient of the curve 𝑦 = 2 + 8𝑥 − 2𝑥 2 at 𝑥 = 3.
2. (a) Find the gradient of the curve 𝑦 = 2𝑥 3 − 15𝑥 2 + 24𝑥 + 6 at:
(i) 𝑥 = 2,
(ii) 𝑥 = 5.
(iii) 𝑥 = 1,
(iv) 𝑥 = 4.
4
3. The equation of a curve is 𝑦 = 𝑥 3 + 𝑥 2 . Find the gradient of the curve at the point with coordinates (2, 9).
16
4. The equation of a curve is 𝑦 = 5𝑥 − 𝑥 2 . Find the gradient of this curve at the point (2, 6).
5. The equation of a curve is 𝑦 = 5𝑥 2 − 3𝑥. Find the gradient of this curve at the point (1, 2).
1
1
6. Calculate the gradient of the curve 𝑦 = 𝑥 3 − 2𝑥 + 𝑥 at the point (2, 4 2).
1
7. Determine the values of 𝑥 for which the gradient of the curve with equation 𝑦 = 3 𝑥 3 − 𝑥 2 − 2𝑥 + 1 is 13.
8. Find the coordinates of the point on the curve with equation 𝑦 = 𝑥 2 + 5𝑥 − 2 at which the gradient is 3.
9. The curve with equation 𝑦 = 𝑎𝑥 2 + 3𝑥 + 2 has a gradient of 10 at the point where 𝑥 = 2. Find the value of
the constant 𝑎.
FAISAL MIZAN
2
10. A curve has equation 𝑦 = 𝑥 3 − 3𝑥 2 + 5𝑥 − 4. (a) Find
𝑑𝑦
.
𝑑𝑥
(b) Find the coordinates of the point on the curve with equation 𝑦 = 𝑥 3 − 3𝑥 2 + 5𝑥 − 4 at which the gradient
is 2.
11. A curve has equation 𝑦 = 6𝑥 2 − 5𝑥 − 12. Find the 𝑥 − coordinate of the point on the curve at which the
gradient is 3.
12. The curve with equation 𝑦 = 5𝑥 − 𝑥 2 intersects the 𝑥 −axis at the origin and at 𝑃.
(a) Write down the 𝑥 −coordinate of 𝑃.
(b) Calculate the coordinates of the point on the curve at which the gradient equals −1.
13. A curve has the equation 𝑦 = 5𝑥 2 − 6𝑥 + 15. Find the 𝑥 coordinate of the point on the curve at which
the gradient of the curve is −2.
Answer:[1] (i) −4, [2] (a) (i) −12, (ii) 24, (iii) 0, (iv) 0, [3] 11, [4] 9, [5] 7, [6] 9.75, [7] 5, −3, [8] (−1, −6),
2
2
[9] 1.75, [10] (a) 3𝑥 2 − 6𝑥 + 5, (b) (1, −1), [11] 3, [12] (a) 5, (b) (3, 6), [13] 𝑥 = 5
∎ Tangnt-Normal:
1. Find the equation of the tangent to the curve at the given value of 𝑥:
(a) 𝑦 = 𝑥 4 − 3 at 𝑥 = 1,
(b) 𝑦 = 𝑥 2 + 3𝑥 + 2 at 𝑥 = −2,
(c) 𝑦 = 2𝑥 3 + 5𝑥 2 − 1 at 𝑥 = 1,
(d) 𝑦 = 5 + 𝑥 at 𝑥 = −2.
2
2. Find the equation of the normal to the curve at the given value of 𝑥:
(a) 𝑦 = 𝑥 2 + 5𝑥 at 𝑥 = −1,
(b) 𝑦 = 3𝑥 2 − 4𝑥 + 1 at 𝑥 = 2,
(c) 𝑦 = 5𝑥 4 − 7𝑥 2 + 2𝑥 at 𝑥 = −1,
(d) 𝑦 = 4 − 𝑥 2 at 𝑥 = −2,
(e) 𝑦 = 2𝑥(𝑥 − 3)3 at 𝑥 = 2,
(f) 𝑦 = 𝑥−2 at 𝑥 = 6.
2
𝑥+2
3. Find the equation of the tangent and the normal to the curve 𝑦 = 𝑥 3 − 2𝑥 2 + 3 at the point 𝑥 = 2
3
4. Find the equation of the tangent and normal to the curve 𝑦 = 5𝑥 − 𝑥 at 𝑥 = 1.
8
5. Find the equation of the normal to the curve 𝑦 = 2𝑥 + 𝑥 at 𝑥 = 1 and 𝑥 = 4 . Find the coordinate of the
point where these normal intersect.
𝑥−3
1
6. Find the equation of the tangent and normal to the curve 𝑦 = 2𝑥−1 when 𝑦 = − 3 .
7. Find the equation of the normal to the curve 𝑦 =
2𝑥−1
√𝑥 2 +5
at the point where 𝑥 = 2 . Give your answer in the
form 𝑎𝑥 + 𝑏𝑦 = 𝑐 , where 𝑎, 𝑏 𝑎𝑛𝑑 𝑐 are integers.
8. The normal to the curve 𝑦 = 2𝑥 2 − 7𝑥 at the point 𝑃 is parallel to the line 2𝑦 + 2𝑥 = 7. Find:
(a) The coordinates of 𝑃,
(b)The equation of the tangent to the curve at 𝑃,
FAISAL MIZAN
3
(c) The coordinates of the point on the curve where the normal cuts the curve again.
9. (a) Find the equation of the tangent to the curve 𝑦 = 3𝑥 2 − 2𝑥 + 5 at 𝑃, which is perpendicular to the line
4𝑦 + 𝑥 = 2.
(b) The tangent meets 𝑥-axis at 𝐴 and 𝑦-axis at 𝐵. Find 𝐴 & 𝐵.
10. (a) Find the equation of the normal to the curve 𝑦 = 3 + 2𝑥 − 𝑥 2 at 𝑄, which is parallel to the line
2𝑦 − 𝑥 = 3.
(b) The normal meets 𝑥-axis at 𝐴 and 𝑦-axis at 𝐵. Find 𝐴 & 𝐵.
11. (a) Find the equation of the tangent to the curve 𝑦 = 𝑥 3 − 6𝑥 2 + 3𝑥 + 10 at the point where 𝑥 = 1.
(b) Find the coordinates of the point where this tangent meets the curve again.
12. A curve has equation 𝑦 = 𝑥 3 − 𝑥 + 6.
(a) Find the equation of the tangent to this curve at the point 𝑃(−1, 6). The tangent at the point 𝑄
is parallel to the tangent at 𝑃.
(b) Find the coordinates of 𝑄,
(c) Find the equation of the normal at 𝑄.
Answers:
1
1
1
[1] (a) 𝑦 = 4𝑥 − 6, (b) 𝑦 = −𝑥 − 2, (c) 𝑦 = 16𝑥 − 10, (d) 𝑦 = − 2 𝑥 + 3. [2] (a) 𝑦 = − 3 𝑥 − 4 3,
1
1
1
3
(b) 𝑦 = − 8 𝑥 + 5 4, (c) 𝑦 = 4 𝑥 − 3 4, (d) 𝑦 = 2𝑥 + 7.5, (e) 𝑦 = −0.1𝑥 − 3.8, (f) 𝑦 = 4𝑥 − 22.
1
[3] 𝑦 = 4𝑥 − 5, 4𝑦 + 𝑥 = 14, [4] 𝑦 = 8𝑥 − 6 , 𝑦 = − 8 𝑥 +
[6] 9𝑦 − 5𝑥 = −13 , 5𝑦 + 9𝑥 =
49
,
3
17
,
8
17
[5] 6𝑦 − 𝑥 = 59 , 3𝑦 + 2𝑥 = 38 , ( 5 ,
52
),
5
[7] 4𝑦 + 9𝑥 = 22, [8](a) (2, −6), (b) 𝑦 = 𝑥 − 8, (c) (1, −5),
1
1
[9](a) 𝑦 = 4𝑥 + 2, (b) 𝐴 (− 2 , 0) , 𝐵(0, 2), [10](a) 𝑦 = 2 𝑥 + 2 (b) 𝐴(−4, 0) , 𝐵(0, 2), [11](a) 𝑦 = −6𝑥 + 14,
1
(b) (4, −10), [12] (a) 𝑦 = 2𝑥 + 8, (b) (1, 6) (c) 𝑦 = − 2 𝑥 +
13
.
2
FAISAL MIZAN
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