Bài tập CALCULUS 04

Created by T Madas Question 12 (**) y = 3x − x − x2 +4, x >0 Find a fully simplified expression for  y dx C1M , x + x3 − x + x −1 + C Question 13 (**) f ( x ) = − x3 + x − 15 x − 13 , x ∈ » a) Find the coordinates of the stationary points of f ( x ) b) Determine the nature of each of the two stationary points found in part (a) c) Hence find the range of values of x for which f ( x ) is decreasing C2C , at (1, −20 ) , max at ( 5,12 ) , x < ∪ x > Created by T Madas Created by T Madas Question 14 (**) y y = x3 − x R2 O R1 x The figure above shows the cubic curve with equation y = x3 − x , x ≥ The curve meets the x axis at the origin O and at the point where x = The finite region R1 is bounded by the curve and the x axis, for ≤ x ≤ The region R2 is bounded by the curve and the x axis, for ≤ x ≤ Show that the area of R1 is equal to the area of R2 C2K , proof Created by T Madas Created by T Madas Question 15 (**) The curve C has equation y= a) Find an expression for x2 + 5x −4, x ≠ dy dx b) Determine an equation of the normal to the curve at the point where x = dy 12 = − , y = 4x − dx x3 Question 16 (**) f ( x) = 6x + x − x2 , x >0 Find a fully simplified expression for  f ( x ) dx C1J , 3x + x + x −1 + C Created by T Madas