Porisma

Worked solution

Transfer an oblique asymptote to the derivative sketch

The diagram shows the graph of y=f(x)y=f(x). Sketch the curve of y=f′(x)y=f^{\prime}(x).

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Given function with asymptotes x=3 and y=−x+2, y-intercept 7/3, minimum (2,1) on the left branch and maximum (4,−3) on the right. The left branch tends up without bound at 3 from the left; the right starts at negative infinity and rises to its maximum before falling.
Given function with asymptotes x=3 and y=−x+2, y-intercept 7/3, minimum (2,1) on the left branch and maximum (4,−3) on the right. The left branch tends up without bound at 3 from the left; the right starts at negative infinity and rises to its maximum before falling.
Given function with asymptotes x=3 and y=−x+2, y-intercept 7/3, minimum (2,1) on the left branch and maximum (4,−3) on the right. The left branch tends up without bound at 3 from the left; the right starts at negative infinity and rises to its maximum before falling.

[2010/VJC/Promo/6(ii)]

CJC · Lecture · Example 7

Read both stationary points

The given minimum (2,1)(2,1) gives derivative zero (2,0)(2,0) and a negative-to-positive crossing. The maximum (4,−3)(4,-3) gives (4,0)(4,0) and a positive-to-negative crossing. Thus the derivative is negative for x<2x<2 and x>4x>4, positive for 2<x<32<x<3 and 3<x<43<x<4, and undefined at 3.

How did this part go?

Retain all asymptotes and branch directions

The Teacher sketch has vertical asymptote x=3x=3, with derivative tending to +∞+\infty from both sides. Both outer tails approach y=−1y=-1 from above, corresponding to the original graph’s limiting slope near y=−x+2y=-x+2. The left derivative branch increases and the right decreases. This uses the slopes of the source sketch; an oblique asymptote alone does not force a derivative limit.

How did this part go?

Keep function heights separate from slopes

The source function has y-intercept 7/37/3, minimum (2,1)(2,1) and maximum (4,−3)(4,-3). These heights do not become derivative heights. No exact formula is supplied. The same attributed question occurs as Review 4, where the source additionally prints 3 marks; no marks are printed in this lecture occurrence.

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Compare your sketch

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Derivative sketch with vertical asymptote x=3 and horizontal asymptote y=−1. Both branches lie above −1, rise to positive infinity near x=3 and cross the x-axis at 2 and 4 respectively. The left branch increases and the right decreases.
Derivative sketch with vertical asymptote x=3 and horizontal asymptote y=−1. Both branches lie above −1, rise to positive infinity near x=3 and cross the x-axis at 2 and 4 respectively. The left branch increases and the right decreases.
Derivative sketch with vertical asymptote x=3 and horizontal asymptote y=−1. Both branches lie above −1, rise to positive infinity near x=3 and cross the x-axis at 2 and 4 respectively. The left branch increases and the right decreases.

Common mistakes

The original y-intercept 7/3 is a height, not a derivative value.

Solution notes

Both source copies visually compared; source wording, displayed results and pagination retained. Independently authored analytical explanations are distinguished from printed working.

Repeated identity compared with cjc-review-differentiation-applications-04: same attribution, labelled graph, ask and answer; lecture number 7 differs from review number 4. Existing canonical review wording and 3 marks preserved. Lecture occurrence retained with its own source pagination, without adding an original-bank identity.

Read the hint