Page 40 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Differentiation
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4. Given the volume of a cone is V = pr h such that r is 2.3 The Second Derivative
2
3
the radius and h is the height of the cone. Determine
(a) dV if h = 2 cm
dr
dh A Determining the second derivative
(b) if r = 5 cm
dV of an algebraic function
5. Determine dy for each of the following functions. 1. Observe that
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dx y = f(x)
1
(a) y = 2x – 1 2 2 dy = f'(x) First derivative
x
2
(b) y = 9x – 1 dx
1 + 3x d y = f"(x)
2
(7 + 3x) 2 dx 2 Second derivative
(c) y =
x 2
2
(d) y = x (5 – x) 2 2. The second derivative, d y 2 is obtained through
2
dx
6. Differentiate each of the following with respect to x. the similar processes of differentiation as in the
1
(a) f(x) = 3 6x – 3 x 2 3 first derivative.
(b) f(x) = 8x – 4 3. dy is the gradient function of the curve y = f(x)
2
(c) f(x) = (3x – 4) 4 dx
5 d y
2
3 at a point (x, y). is the rate of change of the
(d) y = dx 2
6x – 3 gradient of a curve with respect to x.
(e) y = 1
4(5 – 3x) 2
5x
(f) y = – 9
x – 2x 2 2
3
dy Find d y for each of the following functions.
7. Given y = u and u = 5x + 8, find the value of when dx 2
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x = –3. dx (a) y = 3x (b) y = 8(2x – 3)
4
5
dh
8. If h(k – 2k) = 5, find the value of when k = 1.
2
dk Solution
dy (a) y = 3x
5
9. Given y = k(2x – 1) and = 40(2x – 1) , find the
4
n
values of k and n. dx dy = 15x REMEMBER!
4
dx
10. Determine the first derivative for each of the 2 d y
2
following functions: d y = 60x dx 2 can only be
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(a) y = x(2x – 1) dx 2 derived from dy .
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(b) y = (x + 4) 2x – 3 (b) y = 8(2x – 3) dx
4
(c) f(x) = (x – 5) (2x + 7) dy 3
2
3
(d) f(x) = x (9x – 2) 5 dx = 32(2x – 3) (2)
3
(e) f(x) = 3x(1 – x)(5 + x) 4 d y = 64(2x – 3) Form 5
2
1
2
1
(f) f(x) = (2x – 3) – 2x 2 2 dx 2 = 192(2x – 3) (2)
3
x
= 384(2x – 3)
2
11. Differentiate each of the following with respect to x.
5x 2
(a) f(x) = 10
2x – 1
(b) f(x) = √x + 1 Find the value of f"(0) for each of the following
√x – 1 functions.
√2 – x (a) f(x) = x (1 – 3x)
2
(c) f(x) =
x 2 2x – 3x 2
4
(1 – 3x) 4 (b) f(x) =
(d) f(x) = x
x 3
(e) y = 3x – 8x 2 Solution
2
2 + 5x – x 2 (a) f(x) = x (1 – 3x)
x – 2x + 6 = x – 3x
3
2
3
(f) y =
8 – x
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