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Image 1: Graph of a Function with a Tangent Line

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Image 1: Graph of a Function with a Tangent Line

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Vocabulary: function, derivative function, slope, tangent line

Observe this graph closely. What does the tangent line tell you about the function at this point? Describe the connection with derivatives.

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Vocabulary: function, graph, integral, area

Look at the shaded region under the curve. What calculus idea does this image represent? Explain how you would find the area.

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Vocabulary: differential equation, derivative function, critical points

View the slope field shown. What does this pattern say about the possible solutions to the differential equation?

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Vocabulary: secant line, tangent line, derivative, rate of change

This image shows both a secant and a tangent to the same curve. Describe what each line represents in calculus terms.

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Vocabulary: limit, function, asymptote

Notice the approaching value as x moves closer to a point on the function. Describe what a limit is in the context of this image.

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Vocabulary: function, derivative, critical points, maximum, minimum

This diagram sets out a typical optimisation scenario. What steps would you use to find the maximum or minimum value?

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Vocabulary: function, integral, volume, axis of rotation

Visualise the shape formed by rotating a curve around an axis. How do you set up an integral to find its volume?

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Image 7: Second Derivative and Concavity

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Image 7: Second Derivative and Concavity

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Vocabulary: second derivative, concavity, inflection point

This graph shows a function and its points of inflection. How does the second derivative explain the shape of the graph?

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Vocabulary: function, discontinuity, domain, range

Examine the jump in this function. What type of discontinuity is shown? How do you describe it using calculus vocabulary?

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Vocabulary: derivative, composite function, chain rule

Follow how the chain rule is applied to differentiate a composite function. Can you describe what’s happening?

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The tangent line is tangent to the graph of the function y = f(x) at the point (a, f(a)) . It reflects the instantaneous rate of change of the function at this point and geometrically indicates the steepness and trend of change of the function at this point.

The geometric meaning of the derivative f'(a) is the slope m of the tangent line to the function y = f(x) at the point (a, f(a)) , that is, m = f'(a) . The tangent line equation y = mx + b is determined by the derivative. The derivative closely relates the rate of change of the function to the slope of the tangent line and is a key tool for studying the local properties of functions.