Understanding Engineering Mathematics

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10.3.2 Tangent and normal to a curve


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Find the equations of the tangents and the normals to the following curves at the points
indicated.


(i) y=x^2 + 2 x− 3 x= 1
(ii) y=x^4 + 1 x= 1
(iii) y=lnxx= 1
(iv) y=exsinxx= 0

10.3.3 Stationary points and points of inflection


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A.Locate and classify the stationary points and any points of inflection of the following
functions:


(i) x^2 − 4 x+ 3 (ii) x^3 − 12 x+ 2

(iii) x^3 (iv)

x
5

+

5
x
(v) 2x^3 − 15 x^2 + 36 x−4(vi)4x^3 + 3 x^2 − 36 x+ 6

B. Find the maximum and minimum values of the curve of the functiony=x(x^2 − 4 ),
and also find the gradient of the curve at the point of inflection.


10.3.4 Curve sketching in Cartesian coordinates


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Sketch the graphs of the following functions.


(i) x^3 − 2 x^2 −x+ 2 (ii)

x
x+ 1

(iii)

x+ 1
x− 1

(iv)

x− 2
x^2 + 1

(v)

x^2 + 4
x^2 +x− 2

(vi) 3+cos

(x
2

)

(vii) xe−x

10.3.5 Applications of integration – area under a curve


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A.Find the area enclosed between the curve, thex-axis, and the limits stated for each of
the following curves:


(i) y= 2 x^2 +x+ 1 x= 0 , 2 (ii) y=x−

1
x

x= 1 , 2

(iii) (x− 1 )ex x= 1 ,2(iv)y=cos 2xx= 0 ,

π
4
(v) y=x^2 +

1
x^2

x= 1 ,3(vi)y=sin^2 x, x= 0 ,π/ 2
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