10.3.2 Tangent and normal to a curve
➤➤
291 293➤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= 010.3.3 Stationary points and points of inflection
➤➤
291 294➤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+ 6B. 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
➤➤
291 299
➤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−x10.3.5 Applications of integration – area under a curve
➤➤
291 304➤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
xx= 1 , 2(iii) (x− 1 )ex x= 1 ,2(iv)y=cos 2xx= 0 ,π
4
(v) y=x^2 +1
x^2x= 1 ,3(vi)y=sin^2 x, x= 0 ,π/ 2