2.9

Differentiation

9 flashcards to master Differentiation

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Key Concept Flip

What is the derivative of a function f(x), and what does it represent geometrically?

Answer Flip

The derivative, denoted as f'(x) or dy/dx, represents the instantaneous rate of change of the function. Geometrically, it represents the gradient (slope) of the tangent line to the curve of f(x) at a specific point.

Key Concept Flip

Differentiate: y = 3x^4 - 2x^2 + 5x - 7

Answer Flip

Using the power rule (dy/dx(x^n) = nx^(n-1)), dy/dx = 12x^3 - 4x + 5. The derivative gives the gradient function of the original equation.

Key Concept Flip

Explain how to find the equation of the tangent to the curve y = x^2 at the point (2, 4).

Answer Flip

First, find the derivative: dy/dx = 2x. At x = 2, the gradient is 2(2) = 4. Then use the point-gradient form: y - 4 = 4(x - 2) giving y = 4x - 4.

Definition Flip

What is a stationary point, and how do you find it?

Answer Flip

A stationary point is where the gradient of the curve is zero (dy/dx = 0). To find it, differentiate the function, set the derivative equal to zero, and solve for x. Substitute x into the original equation to find the y-coordinate.

Key Concept Flip

How do you determine whether a stationary point is a maximum or minimum point?

Answer Flip

Use the second derivative test. If the second derivative (d^2y/dx^2) is positive at the stationary point, it's a minimum. If it's negative, it's a maximum. If it's zero, the test is inconclusive.

Key Concept Flip

Find the coordinates of the stationary points of the curve y = x^3 - 3x

Answer Flip

dy/dx = 3x^2 - 3. Set dy/dx = 0: 3x^2 - 3 = 0 => x = ±1. When x = 1, y = -2; when x = -1, y = 2. Stationary points are (1, -2) and (-1, 2).

Key Concept Flip

A particle moves along a line such that its displacement s (in meters) from a fixed point O at time t (in seconds) is given by s = t^3 - 6t^2 + 9t. Find the times when the particle is instantaneously at rest.

Answer Flip

The particle is at rest when its velocity is zero. Velocity, v = ds/dt = 3t^2 - 12t + 9. Setting v = 0: 3t^2 - 12t + 9 = 0 => t = 1 and t = 3 seconds.

Key Concept Flip

Explain the meaning of 'rate of change' in the context of differentiation.

Answer Flip

Rate of change describes how one variable changes in relation to another. In differentiation, dy/dx represents the rate of change of y with respect to x.

Example: ds/dt is the rate of change of displacement (s) with respect to time (t) - i.e., velocity.
Key Concept Flip

Find the maximum value of the function f(x) = -x^2 + 4x + 3.

Answer Flip

f'(x) = -2x + 4. Setting f'(x) = 0 gives x = 2. f''(x) = -2, which is negative, so x=2 is a maximum. f(2) = -4 + 8 + 3 = 7. Maximum value is 7.

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2.8 Functions 3.1 Coordinates

Key Questions: Differentiation

What is a stationary point, and how do you find it?

A stationary point is where the gradient of the curve is zero (dy/dx = 0). To find it, differentiate the function, set the derivative equal to zero, and solve for x. Substitute x into the original equation to find the y-coordinate.

About Differentiation (2.9)

These 9 flashcards cover everything you need to know about Differentiation for your Cambridge IGCSE Mathematics (0580) exam. Each card is designed based on the official syllabus requirements.

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