Exam-Style Questions.

Some of the key features of the following graph can be found by using differentiation:

(a) When \( x = 3 \) find the value of \( \dfrac{dy}{dx} \)

(b) Find the coordinates of the point on the graph of \( y = 4x^2 - 5x + 9 \) where the gradient is zero.

(c) When \( y = 5x^p + qx^3 \), \( \quad \dfrac{dy}{dx} = 35x^6 + 30x^2 \)

Find the value of \( p \) and the value of \( q \).

Worked Solution

The acceleration, \(a\) ms^-2 , of an object at time \(t\) seconds is given by

The object is at rest when \(t=1\).

Find the velocity of the object when \(t=7\).

Worked Solution

(a) Find \(\int (4x+5) dx\).

(b) Given \(f'(x) = 4x+5\) find \(f(x)\) if \(f(3.4) = 10.12\).

Worked Solution

(a) Find \( \frac{dy}{dx} \) when:

(b) Find the following integral:

Worked Solution

The function \(f\) is defined by \(f(x) = 8 - 5 \sin{x} \), for \( x \ge 0 \).

The diagram shows part of the graph of \(y = f (x) \).

The shaded region is enclosed by the graph of \(y=f(x)\) and the x-axis for the first three periods of the function.

(a) Find the exact value of the x-coordinate of the right side of the shaded region.

(b) Show that the area of the shaded region is \( 48 \pi \).

A hemisphere has a total surface area in square centimetres equal to the shaded area in the previous diagram.

(c) Find the radius of the hemisphere.

Worked Solution

The following diagram shows part of the graph of:

The shaded region A is bounded by the x-axis, y-axis and the graph of \(f\).

(a) Write down an integral for the area of region A.

(b) Find the area of region A.

The three points A (0 , 0) , B (6 , 9) and C (p , 0) define the vertices of a triangle.

(c) Find the value of p, the x-coordinate of C , such that the area of the triangle is equal to the area of region A.

Worked Solution

Consider the graph of the function \(f(x)=x^2+2\).

(a) Find the area between the graph of \(f\) and the x-axis for \(2\le x \le 3\).

(b) If the area described above is rotated 360^o around the x-axis find the volume of the solid formed.

Worked Solution

This graph represents the function \(f:x\to a \cos x, a\in \mathbf N\)

(a) Find the value of \(a\).

(b) Find the area of the shaded region.

Worked Solution

The diagram shows a sketch of the curve C with equation:

The point P with coordinates (4, 3) lies on C.

The line L is the tangent to C at the point P.

The region A, shown shaded, is bounded by the curve C, the line L and the y-axis.

Find the area of A making your method clear.

Note that solutions based entirely on graphical or numerical methods are not acceptable.)

Worked Solution

Let \(f(x) = \frac{ln3x}{kx} \) where \( x \gt 0\) and \( k \in \mathbf Q^+ \).

(a) Find an expression for the first derivative \(f'(x) \).

The graph of \(f\) has exactly one maximum point at P.

(b) Find the x-coordinate of P.

The graph of \(f\) has exactly one point of inflection at Q.

(c) Find the x-coordinate of Q.

(d) The region enclosed by the graph of \(f\), the x-axis, and the vertical lines through P and Q has an area of one square unit, find the value of \(k\).

Worked Solution