Exam-Style Questions.

1.	IB Standard

Let \(f(x)=\frac{2x}{x^2+3}\)

(a) Use the quotient rule to show that \(f'(x)=\frac{6-2x^2}{(x^2+3)^2}\).

(b) Find \(\int \frac{2x}{x^2+3}dx\).

(c) The area enclosed by the graph of \(f(x)\), the x-axis and the lines \(x=\sqrt3\) and \(x=n\) has an area of \(\ln14\). Find the value of \(n\).

Worked Solution

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2.	IB Standard

Very accurate equipment was used to measure the movement of a particle which moved in a straight line for 3 seconds. Its velocity, \(v\) ms^-1 , at time \(t\) seconds, is given by:

$$v=(t^2-5)^3$$

(a) Find the velocity of the particle when \(t=2\).

(b) Find the value of t for which the particle is at rest.

(c) Find the total distance the particle travels during the first three seconds.

(d) Show that the acceleration of the particle is given by \(a=6t(t^2-5)^2\)

(e) Find all possible values of t for which the velocity and acceleration are both positive or both negative.

Worked Solution

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3.	IB Analysis and Approaches

Consider the function \(f\) defined by \(f(x)= \ln{(x^2 - 9)}\) for \(x > 3\).

The following diagram shows part of the graph of \(f\) which crosses the x-axis at point \(A\) with coordinates \((a,0)\).

The line \(L\) is the tangent to the graph of \(f\) at the point \(B\) with coordinates \((b,c)\). The gradient of \(L\) is \( \frac14\)

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

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

Worked Solution

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4.	IB Standard

If \(f(x)=x\sin{x}\), for \(-3\le x\le3\)

(a) Find \(f'(x)\)

(b) Sketch the graph of \(y=f'(x)\), for \(-3\le x\le3\)

Worked Solution

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5.	IB Analysis and Approaches

A particle moves in a straight line such that its velocity, \(v\) ms^-1, at time \(t\) seconds is given by:

$$ v(t)=10e^{-\frac{t}{7}} \sin\left(\frac{t}{2}\right) $$

for \( 0 \le t \le 4 \pi\). The graph of \(v\) is shown in the following diagram.

Let \(t_1 \) be the first time when the particle's acceleration is zero.

(a) Find the value of \(t_1\).

(b) Find the distance travelled by the particle between \(t = t_1 \), and t = \(4 \pi \).

Worked Solution

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6.	IB Standard

Let \(f(x)=\frac{g(x)}{h(x)}\), where \(g(3)=36\), \(h(3)=12\), \(g'(3)=10\) and \(h'(3)=4\). Find the equation of the normal to the graph of \(f\) at \(x=3\).

Worked Solution

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7.	IB Analysis and Approaches

The displacement, in millimetres, of a particle from an origin, O, at time t seconds, is given by \(s(t) = t^3 cos t + 5t sin t\) where \( 0 \le t \le 5 \) .

(a) Find the maximum distance of the particle from O.

(b) Find the acceleration of the particle at the instant it first changes direction.

Worked Solution

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8.	IB Analysis and Approaches

The edge lengths, \(x\) cm, of a cube are increasing at a rate of 6 cm s^-1.

Find the rate at which the volume of the cube, \(V\) cm³, is increasing when the edge lengths are 20cm.

Worked Solution

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9.	IB Standard

The function \(f\) is defined as follows:

$$f(x)=\frac{122}{1+60e^{-0.3x}}$$

(a) Calculate \(f(0)\).

(b) Find a value of \(x\) for which \(f(x)=85\)

(c) Find the range of \(f\).

(d) Show that:

$$f'(x)=\frac{2196e^{-0.3x}}{(1+60e^{-0.3x})^2}$$

(e) Find the maximum rate of change of \(f\).

Worked Solution

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10.	IB Standard

Make a sketch of a graph showing the velocity (in \(ms^{-1}\)) against time of a particle travelling for six seconds according to the equation:

$$v=e^{\sin t}-1$$

(a) Find the point at which the graph crosses the \(t\) axis.

(b) How far does the particle travel during these first six seconds?

Worked Solution

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11.	IB Analysis and Approaches

The function \(f\) is such that \(f(x) = \frac{\ln2x}{x^3} \) where \(x \gt 0\).

(a) Find the first derivative of the function, \( f'(x) \).

(b) The graph of \( y=f(x) \) has a horizontal tangent at the point T. Find the coordinates of T.

(c) Show that T is a local maximum point by considering the second derivative, \( f''(x) \).

(d) Find the values of \(x\) for which \( f(x) \gt 0 \).

(e) Sketch the graph of \( f \) showing clearly the value of the x-intercept and the approximate position of point T.

Worked Solution

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12.	IB Analysis and Approaches

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

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