Gradient of tangents on curves

Task goals

• Estimate the gradient of a curve at a stated x-value by reading the rise and run of the tangent line already drawn on the graph.
• Recognise that the tangent at a turning point is horizontal and state that the gradient there is zero.
• Compare the tangent gradients at two different points on a curve and say which is steeper, which is more positive, and which is more negative.
• Sketch an approximate tangent at a marked point on a quadratic or cubic and estimate its gradient from that sketch.
• State the intervals of x on which a graphed function is increasing or decreasing, equivalently where f'(x) > 0 and where f'(x) < 0.
• Read from a graph the x-values at which the gradient takes a given value, including listing every x-value where f'(x) = 0.
• Locate all stationary points on a graphed curve, state the gradient at each, and classify each as a maximum, a minimum, or a point of inflection.
• Describe in words or with a gradient sign diagram how the gradient of a curve changes across its domain, and explain why the gradient differs between two given points.