Exercises : Application of differentiation & Application of integration

Application of differentiation

1.    Determine the intervals of y = x³ + x² – 5x – 5 where it is increasing and decreasing. Hence, sketch the graph.

2.    Find the stationary points and state the nature of the curve y = xe^-x . Hence, show that the curve has a point of inflection at x = 2 and sketch its graph.

3.   Find the maximum and minimum points of the curve 

4.    Find the coordinates of turning point of the curve y = 2x^2x + 8e^-3x and state its nature.

5.    Determine the stationary point of the curve y = x – In (1 + x) and sketch its graph.

Application of integration

1.    Sketch and find the area of the region between the curve y = x³ and the line y = -x and y =1.

2.    Find the area of the region between the x-axis and y = (x - 1)³ from x=0 to x=23

3.    Find the volume of the solid generated by revolving the region enclosed by y²=x and y = -x + 2 about the indicated line y-axis.

4.    Find the volume of the solid generated by revolving the region enclosed by y = x² + 1 and y = x +3 about the indicated line x-axis.

5.    Find the volume of the solid generated by revolving the region enclosed by y = x and y = x² about the indicated line y-axis.