Page 41 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Differentiation
f'(x) = 2x – 9x
2
f"(x) = 2 – 18x 2.4 Application of Differentiation
f"(0) = 2 – 18(0)
= 2
A Interpreting the gradient of tangent
4
(b) f(x) = 2x – 3x 2 to a curve at different points
x
= 2x – 3x 1. The first derivation of a function y = f(x), that is
3
f'(x) = 6x – 3 dy = f'(x) is the gradient function which gives
2
dx
f"(x) = 12x us the value of the gradient of the tangent to the
f"(0) = 0 curve y = f(x) at a point on the curve.
Try Questions 1 – 7 in ‘Try This! 2.3’ xReserved.
y
A
Try This! 2.3
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y = f(x)
2
1. Determine dy and d y 2 for each of the following 0 tangent m = dy
functions. dx dx dx
(a) y = 8x (b) y = 7x – 8x 2. The gradient of the tangent line drawn at a point
4
3
2
x
(c) y = – √x (d) y = x (1 – 2x) A(x , y ) on the curve is obtained by substituting
3
2 1 1 dy
(e) y = 5(4x – 3) (f) y = 6x – 5x 3 the value of x into = f(x).
6
x 2 1 dx
2. Find f '(x) and f"(x) for each of the following functions. 3. The accuracy of the value of gradient m, can

1
2
(a) f(x) = 7x – 5x + be tested/verified by drawing a graph manually
x
(b) f(x) = 4x (x + 2) on a graph paper or by using computer apps/
2
2
(c) f(x) = 6 programs.
x + 3
3x
(d) f(x) =
1 – x 11
3. Find the value of f"(0) for each of the following.
3
4
(a) f(x) = (9 – 3x) Find the gradient of the tangent to the curve y = 4x – 2
(b) f(x) = 4x(x – 3) at the given points.
2
(c) f(x) = x + 3 (a) Point A(1, 2)
x – 3
4. Given the function f(x) = 3x + mx – 4. Find (b) Point B 1 1 , – 3 2
5
3
(a) the second derivative of the function f. 2 2
(b) the value of m if f"(–1) = –5.
Solution
5. Given the function y = (x – 3) . 3 3
2
dy d y (a) y = 4x – 2 (b) y = 4x – 2
2
(b) Find and . dy dy
2
2
dx dx 2 = 12x = 12x
Form 5
2
(c) Show that 2y – x dy + (3x – 9) d y = 0. dx = 12(1) 2 dx 1 2
dx dx 2 = 12 1 2
= 12 2
6. Given the function y = x(4 – x).
1
1 dy d y = 12 1 2
2
(a) Show that y – x + x = 0. 4
2 dx dx 2
(b) Find the possible values of x when = 3
2
y = x dy + 2x d y .
dx dx 2 12
7. Given the function y = 3(x – 1) . Calculate the gradient of the tangent to the curve
2
2
(a) Express y d y + dy in terms of x.
dx 2 dx y = 6 + 2x at the point where the y-coordinate is –2.
2
(b) Solve the equation y d y + dy = 0. x
dx 2 dx
250