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Applications of differentiation

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Questions 1–6 in this collection

Practice questions

Worked example · 4 of 6

Read stationary points from a derivative graph

The derivative graph y=f′(x)y=f^{\prime}(x) is shown. Find the x-coordinates of the stationary points of f(x)f(x) and determine their nature.

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Illustrative upward-opening derivative graph with zeros 0 and 2, positive outside(0,2), negative inside. Vertical scale is illustrative; no exact derivative formula is asserted.
Illustrative upward-opening derivative graph with zeros 0 and 2, positive outside(0,2), negative inside. Vertical scale is illustrative; no exact derivative formula is asserted.
Illustrative upward-opening derivative graph with zeros 0 and 2, positive outside(0,2), negative inside. Vertical scale is illustrative; no exact derivative formula is asserted.

CJC · Lecture · Example 3

Solution

Worked example · 5 of 6

Match slope and concavity intervals on a cubic graph

Using the given graph of y=0.1x(x−6)(x+6)y=0.1x(x-6)(x+6), match (a) strictly increasing, (b) strictly decreasing, (c) concave upwards and (d) concave downwards to the printed sets: {x∈R:−3.46<x<3.46}\{x\in\mathbb R:-3.46<x<3.46\}; {x∈R:x<0}\{x\in\mathbb R:x<0\}; {x∈R:x>0}\{x\in\mathbb R:x>0\}; {x∈R:x<−3.46 or x>3.46}\{x\in\mathbb R:x<-3.46\text{ or }x>3.46\}.

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Graph of 0.1x(x−6)(x+6), roots −6,0,6, printed approximate turning labels (−3.46,8.31),(3.46,−8.31). Concavity changes at 0; turning abscissae exact ±2sqrt3.
Graph of 0.1x(x−6)(x+6), roots −6,0,6, printed approximate turning labels (−3.46,8.31),(3.46,−8.31). Concavity changes at 0; turning abscissae exact ±2sqrt3.
Graph of 0.1x(x−6)(x+6), roots −6,0,6, printed approximate turning labels (−3.46,8.31),(3.46,−8.31). Concavity changes at 0; turning abscissae exact ±2sqrt3.

CJC · Lecture · Example 4

Solution

Worked example · 6 of 6

Sketch a derivative from a smooth peak and horizontal tails

Sketch the graph of the derivative function for the given graph.

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Given qualitative graph: a smooth maximum at (0,2), rising for negative x and falling for positive x, with horizontal asymptote y=1 at both ends. No exact formula is asserted.
Given qualitative graph: a smooth maximum at (0,2), rising for negative x and falling for positive x, with horizontal asymptote y=1 at both ends. No exact formula is asserted.
Given qualitative graph: a smooth maximum at (0,2), rising for negative x and falling for positive x, with horizontal asymptote y=1 at both ends. No exact formula is asserted.

[2006/HCI/Promo/1]

CJC · Lecture · Example 5

Solution

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