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SQA Higher Maths Differentiation and graphs

Differentiation and graphs

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The gradient of a curve is given by the derivative of the equation of the curve.

Gradient of a curve

The second derivative of a function f(x), given by f''(x), and the second derivative of an equation y, given by dΒ²y/dxΒ², can be found by differentiating the function and the equation twice. For graphs, the second derivative represents the rate of change of the gradient.

Second derivative

At the stationary points and points of inflection of a graph, the derivative of the function of the equation of the graph is 0. Therefore, these points can be found by finding the derivative of the function, equating it to 0 and solving for x.

Stationary points and points of inflection

The maxima of a graph are the high points of the graph at which the slope is zero. At maxima, the second derivative of the function is less than 0. The minima of a graph are the low points of the graph at which the slope is zero. At minima, the second derivative of the function is more than 0.

Maxima and minima

A function f(x) is said to be increasing at point x if f'(x) > 0 or decreasing at point x if f'(x) < 0.

Increasing and decreasing functions

1

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Given that the equation of a curve is y = x(cosx +xΒ²), find the gradient of the curve.

SQA Higher Maths Differentiation and graphs Given that the equation of a curve is y = x(cosx +xΒ²), find the gradient of the curve.
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2

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Find the gradient of the curve y = 1/(xΒ³ + 18x) at x = 1.

SQA Higher Maths Differentiation and graphs Find the gradient of the curve y = 1/(xΒ³ + 18x) at x = 1.
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3

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How is the second derivative of a function f(x) found and how is it denoted?

SQA Higher Maths Differentiation and graphs How is the second derivative of a function f(x) found and how is it denoted?
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4

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Find the rate of change of the gradient of the graph y = 18xsinx.

SQA Higher Maths Differentiation and graphs Find the rate of change of the gradient of the graph y = 18xsinx.
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5

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Find the second derivative of the function f(x) = (cosΒ²x + 7x)Β².

SQA Higher Maths Differentiation and graphs Find the second derivative of the function f(x) = (cosΒ²x + 7x)Β².
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6

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Use differentiation to show that the graph given by the equation y = 5/(x + 17) does not have any stationary or inflection points.

SQA Higher Maths Differentiation and graphs Use differentiation to show that the graph given by the equation y = 5/(x + 17) does not have any stationary or inflection points.
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7

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Is (2, 125) a stationary point of the graph y = (1 + xΒ²)Β³?

SQA Higher Maths Differentiation and graphs Is (2, 125) a stationary point of the graph y = (1 + xΒ²)Β³?
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8

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Check whether (1, 19) is a stationary point of the graph given by the equation y = xΒ³ + 8x + 10.

SQA Higher Maths Differentiation and graphs Check whether (1, 19) is a stationary point of the graph given by the equation y = xΒ³ + 8x + 10.
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9

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Given that (2, 10) is a minimum point of a graph given by the function f(x), evaluate f'(2).

SQA Higher Maths Differentiation and graphs Given that (2, 10) is a minimum point of a graph given by the function f(x), evaluate f'(2).
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10

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Find the stationary points of y = xΒ³/3 βˆ’ xΒ²/2 βˆ’ 6x and classify them as maxima and minima.

SQA Higher Maths Differentiation and graphs Find the stationary points of y = xΒ³/3 βˆ’ xΒ²/2 βˆ’ 6x and classify them as maxima and minima.
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11

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Find whether the curve y = 4xΒ² βˆ’ xΒ³ at x = 3 Is increasing or decreasing.

SQA Higher Maths Differentiation and graphs Find whether the curve y = 4xΒ² βˆ’ xΒ³ at x = 3 Is increasing or decreasing.
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12

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Is the function f(x) = tanx + 5xΒ³ increasing or decreasing at x = βˆ’3?

SQA Higher Maths Differentiation and graphs Is the function f(x) = tanx + 5xΒ³ increasing or decreasing at x = βˆ’3?
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13

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What is the interval at which y = βˆ’xΒ³ + 12x + 50 is decreasing?

SQA Higher Maths Differentiation and graphs What is the interval at which y = βˆ’xΒ³ + 12x + 50 is decreasing?
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