# Mercator chart

Classified in Design and Engineering

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Unit 6: Curve Sketching

Ex 1) Analyze & Sketch f(x) = 2x3-3x2-3x+2

Domain: D={XER}                                -1| 2, -3, -3, 2

Intercepts: 0=2³-3²-3+2                                    -2, 5, 2

0=0             (x+1)                         2, -5, 2, 0

Y-int:                                                  (x+1)(2x²-5x+2)=0

f(0) = 2                                               (x+1)(x-2)(2x+1)=0

(0,2)                                                 X = -1, 2, 1/2

2x2-4x-x-12

int chart                                =2x(x-2)-(x-2)

=(x-2) (2x-1)

Inc/dec:                                                x=6+-√(-6)2-4(6)(-3)

y'=6x2-6x-3                                                       2(6)

x=6+-√108

Concavity: Y''=12x-6                                     12

-12x=-6                                 x=1.4,-0.4

x=0.5                 f(1.4)=-2.6            f(-0.4)=2.6

(0.5, 0)                  (1.4,-2.6)               (-0.4,2.6)

min                      max

Ex 2) Open topped box base twice as long as width. Volume 2400cm3 find dimension minimize amount of cardboard required. (SA)

SA = 2x(x)+2(h)(x)+2(2x)(h)

= 2x² + 2hx + 4xh                                                             Dimensions; W=12.16cm, L=24.4cm, H=8.1cm

=2x² + 6hx

=2x² + 6x (1200/x²)                    2x(x)h=2400

=2x² + 7200^x-1                         2x^2h=2400

SA'=4x-7200x^-2                              h=1200/x^2

0=4x-7200x^-2                            h=1200/12.16^2

7200/x^2 =4x                               =8.1cm

7200=4x^3

3√7200 = x

4

12.16 = x