Porisma

Worked solution

Sketch a derivative across two vertical asymptotes

The diagram, not to scale, shows the graph of y=f(x)y=f(x). The vertical asymptotes are x=0x=0 and x=2x=2. Sketch the graph of y=f′(x)y=f^{\prime}(x), showing all corresponding coordinates and asymptotes where possible. marks

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Qualitative graph, not to scale: local maximum A(-1,2) on the negative-x branch, zero at -0.5, vertical asymptote x=0; between 0 and 2 a U-shaped branch has minimum B(1,3); beyond x=2 the curve rises from negative infinity towards the x-axis. No numerical function is supplied.
Qualitative graph, not to scale: local maximum A(-1,2) on the negative-x branch, zero at -0.5, vertical asymptote x=0; between 0 and 2 a U-shaped branch has minimum B(1,3); beyond x=2 the curve rises from negative infinity towards the x-axis. No numerical function is supplied.
Qualitative graph, not to scale: local maximum A(-1,2) on the negative-x branch, zero at -0.5, vertical asymptote x=0; between 0 and 2 a U-shaped branch has minimum B(1,3); beyond x=2 the curve rises from negative infinity towards the x-axis. No numerical function is supplied.

[2010/SRJC/I/6(a)]

CJC · Review problems · Question 1

Answer

Zeros (−1,0)(-1,0) and (1,0)(1,0); source answer has asymptotes x=0x=0, x=2x=2 with the branches shown.

Translate stationary points into zeros

The source curve has a local maximum A(−1,2)A(-1,2) and local minimum B(1,3)B(1,3). Its derivative crosses the axis at (−1,0)(-1,0) from positive to negative and at (1,0)(1,0) from negative to positive. The original zero x=−0.5x=-0.5 does not specify a derivative zero.

How did this part go?

Retain the source asymptotic branches

The supplied answer sketch has vertical asymptotes x=0x=0 and x=2x=2: its slopes tend to negative infinity on both sides of 0 and positive infinity on both sides of 2. Its outer derivative branches approach the x-axis. The three branches in the reconstructed answer retain those behaviours and both labelled zeros.

How did this part go?

Read what is and is not determined

The derivative is positive for x<−1x<-1, negative for −1<x<0-1<x<0 and 0<x<10<x<1, and positive for 1<x<21<x<2 and x>2x>2, following the pictured rises and falls. These are the qualitative features requested by the source sketch, not a uniquely determined formula. A vertical asymptote of a general function alone does not prove a derivative asymptote; here the branch shape is also part of the supplied schematic.

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Compare your sketch

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Qualitative derivative has zeros A(-1,0),B(1,0). It is positive left of -1, negative between -1 and 0 and between 0 and 1, positive between 1 and 2 and right of 2. Both sides at x=0 tend down without bound; both sides at x=2 tend up without bound. Outer branches approach y=0. No unique derivative formula is asserted.
Qualitative derivative has zeros A(-1,0),B(1,0). It is positive left of -1, negative between -1 and 0 and between 0 and 1, positive between 1 and 2 and right of 2. Both sides at x=0 tend down without bound; both sides at x=2 tend up without bound. Outer branches approach y=0. No unique derivative formula is asserted.
Qualitative derivative has zeros A(-1,0),B(1,0). It is positive left of -1, negative between -1 and 0 and between 0 and 1, positive between 1 and 2 and right of 2. Both sides at x=0 tend down without bound; both sides at x=2 tend up without bound. Outer branches approach y=0. No unique derivative formula is asserted.

Common mistakes

The original graph’s y-coordinates 2 and 3 become zero slopes, not derivative y-values.

Solution notes

Both source prompt and answer figures independently reconstructed. No unique derivative formula or extra turning-point coordinates are invented. Source asymptotic slope behaviour is qualified as information in the supplied schematic.

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