User interface language: English | Español

Date November 2021 Marks available 1 Reference code 21N.1.AHL.TZ0.8
Level Additional Higher Level Paper Paper 1 Time zone Time zone 0
Command term Interpret Question number 8 Adapted from N/A

Question

Juri skis from the top of a hill to a finishing point at the bottom of the hill. She takes the shortest route, heading directly to the finishing point (F).

Let h(x) define the height of the hill above F at a horizontal distance x from the starting point at the top of the hill.

The graph of the derivative of h(x) is shown below. The graph of h(x) has local minima and maxima when x is equal to a, c and e. The graph of h(x) intersects the x-axis when x is equal to b, d, and f.

Identify the x value of the point where |h(x)| has its maximum value.

[1]
a.i.

Interpret this point in the given context.

[1]
a.ii.

Juri starts at a height of 60 metres and finishes at F, where x=f.

Sketch a possible diagram of the hill on the following pair of coordinate axes.

[3]
b.

Markscheme

a             A1


[1 mark]

a.i.

the hill is at its steepest / largest slope of hill              A1


[1 mark]

a.ii.

            A1A1A1


Note:
Award (A1) for decreasing function from 0 to b and d to f and increasing from b to d; (A1) for minimum at b and max at d; (A1) for starting at height of 60 and finishing at a height of 0 at f. If reasonable curvature not evident on graph (i.e. only straight lines used) award A1A0A1.


[3 marks]

b.

Examiners report

This was one of the weakest questions on the paper. Many candidates failed to appreciate the significance of the absolute value and gave c as the maximum value rather than a. Another common error was to interpret the maximum value as greatest velocity or highest point rather than the point where the hill was steepest. A few candidates drew a graph that went from the starting point to the finishing point. What happened in between, often, showed little understanding of the relationship between the graphs of a function and its derivative. The section of the syllabus that mentions understanding derivatives through graphical methods needs more support from teachers.

a.i.

This was one of the weakest questions on the paper. Many candidates failed to appreciate the significance of the absolute value and gave c as the maximum value rather than a. Another common error was to interpret the maximum value as greatest velocity or highest point rather than the point where the hill was steepest. A few candidates drew a graph that went from the starting point to the finishing point. What happened in between, often, showed little understanding of the relationship between the graphs of a function and its derivative. The section of the syllabus that mentions understanding derivatives through graphical methods needs more support from teachers.

a.ii.
[N/A]
b.

Syllabus sections

Topic 5—Calculus » SL 5.1—Introduction of differential calculus
Show 80 related questions
Topic 5—Calculus » SL 5.2—Increasing and decreasing functions
Topic 5—Calculus » SL 5.6—Stationary points, local max and min
Topic 5—Calculus » AHL 5.13—Kinematic problems
Topic 5—Calculus

View options