Product rule

The diagram shows the curve with equation y = (x³ - 2x²)ln x. The curve has a point of inflection at the point M.

Show that the x-coordinate of M satisfies the equation

Use an iterative formula, based on the equation in part (a)(i), to determine the x-coordinate of M correct to 2 decimal places. Use an initial value of 1.1 and show the

result of each step of the iterative process.

Determine the exact area of the shaded region, giving your answer in the form p ln q - r,

where p and r are positive rational numbers and q is a positive integer.

Determine the x-coordinates of any stationary points on the curve.

Show that the curve is convex for all values of x.

The curve y = (x² - 2)ln x has one stationary point which is close to x = 1.

Show that the x-coordinate of this stationary point satisfies the equation 2x² ln x + x² - 2 = 0

Show that the Newton-Raphson iterative formula for finding the root of the equation in part (a)

Apply the Newton-Raphson formula with initial value x₁ = 1 to find x₂ and x₃.

Find the coordinates of this stationary point, giving each coordinate correct to 3 decimal

places.

The diagram shows the curve M with equation y = xe^-2x.

Show that M has a point of inflection at the point P where x = 1.

The line L passes through the origin O and the point P. The shaded region R is enclosed by the curve M and the line L.

Show that the area of R is given by

where a and b are integers to be determined.

Differentiate the following with respect to x.

(2x + 3)⁷

x³ ln x

Find the equation of the curve through (1, 3) for which = 6x - 5.

The diagram shows the curve C with parametric equations

The maximum point on C is denoted by P.

Determine the exact coordinates of P.

The shaded region R is enclosed by the curve, the x-axis and the lines x = 1 and x = 6.

Show that the area of R is given by

where a and b are constants to be determined.

Hence determine the exact area of R.

By considering show that the curve is convex for all values of x.

The diagram shows the part of the curve y = 3x sin 2x for which 0 ⩽ x ⩽ .

The maximum point on the curve is denoted by P.

Show that the x-coordinate of P satisfies the equation tan 2 x + 2 x = 0.

Use the Newton-Raphson method, with a suitable initial value, to find the x-coordinate of P,

giving your answer correct to 4 decimal places. Show the result of each iteration.

The trapezium rule, with four strips of equal width, is used to find an approximation to

Explain briefly why it is not easy to tell from the diagram alone whether the trapezium rule with four strips of equal width gives an under-or over-estimate for the area of the

region in this case.