Gradient of a Chord and the Limit as h approaches 0 (Grade 12)
Free printable Grade 12 Mathematical Methods worksheet: the gradient of a curve as the limit of chord gradients (first-principles intuition for the derivative). Every value computed.
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Grade 12 · Math worksheet
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Differentiation: Gradient of a Chord and the Limit as h approaches 0
Find the gradient of each chord (average rate of change), then find the value the chord gradient approaches as h shrinks to 0. This limit is the gradient of the curve, f'(x), at that point.
- 1.For f(x) = 4x^2 - 4x + 1, find the gradient of the chord (average rate of change) between x = 0 and x = 1.
- 2.For f(x) = -3x^2 + x - 1, find the gradient of the chord (average rate of change) between x = -1 and x = 2.
- 3.For f(x) = -4x^2 + 6x - 4, find the gradient of the chord (average rate of change) between x = 3 and x = 6.
- 4.For f(x) = - x - 5, find the gradient of the chord (average rate of change) between x = 1 and x = 4.
- 5.For f(x) = 4x^2 + 6x - 6, the gradient of the chord from x = 2 to x = 2 + h is calculated for shrinking h below. What value does it approach as h -> 0? (This limit is f'(2).)
- 6.For f(x) = -4x^2 + 3x - 4, the gradient of the chord from x = -1 to x = -1 + h is calculated for shrinking h below. What value does it approach as h -> 0? (This limit is f'(-1).)
- 7.For f(x) = -3x^2 + 3x - 1, the gradient of the chord from x = 2 to x = 2 + h is calculated for shrinking h below. What value does it approach as h -> 0? (This limit is f'(2).)
- 8.For f(x) = 3x^2 - 2x - 6, the gradient of the chord from x = 3 to x = 3 + h is calculated for shrinking h below. What value does it approach as h -> 0? (This limit is f'(3).)
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