Exam-Style Questions.

1.	IB Studies

A function is given as \(f(x)=3x^2-6x+4+\frac3x,-2\le x \le 4, x\ne 0\).

(a) Find the derivative of the function.

(b) Find the coordinates of the local minimum point of \(f(x)\) in the given domain using your calculator.

Worked Solution

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2.	IB Studies

Consider the graph of the function \(f(x)=7-3x^2-x^3\)

(a) Label the local maximum as A on the graph.

(b) Label the local minimum as B on the graph.

(c) Write down the interval where \(f(x)>5\).

(d) Draw the tangent to the curve at \(x=-3\) on the graph.

(e) Write down the equation of the tangent at \(x=-3\).

Worked Solution

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3.	IB Studies

A package is in the shape of a cuboid and has a length \(l\) cm, width \(w\) cm and height of 12 cm.

(a) Express the volume of the package in terms of \(l\) and \(w\).

The total volume of the package is 2400 cm³.

(b) Show that \(l=\frac{200}{w}\).

The package is tied up using a length of red string that fits exactly around the package in two different directions, as shown in the following diagram (not to scale).

(c) Show that the length of string, \(x\)cm, required to tie up the package can be written as \(24+4w+\frac{400}{w}\)

(d) Sketch the graph of \(x\) for \(0\lt w \le 12\), clearly showing the local minimum point.

(e) Find \(\frac{dx}{dw}\).

(f) Find the value of \(w\) for which \(x\) is a minimum.

(g) Find the value, \(l\), of the package for which the length of string is a minimum.

(h) Find the minimum length of string required to tie up the package.

Worked Solution

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4.	IB Studies

Consider the function \(f(x)=x^3-9x+2\).

(a) Sketch the graph of \(y=f(x)\) for \(-4\le x\le 4\) and \(-14\le y\le 14\) showing clearly the axes intercepts and local maximum and minimum points. Use a scale of 2 cm to represent 1 unit on the x-axis, and a scale of 1 cm to represent 2 units on the y-axis.

(b) Find the value of \(f(-1)\).

(c) Write down the coordinates of the y-intercept of the graph of \(f(x)\).

(d) Find \(f'(x)\).

(e) Find \(f'(-1)\)

(f) Explain what \(f'(-1)\) represents.

(g) Find the equation of the tangent to the graph of \(f(x)\) at the point where x is –1.

R and S are points on the curve such that the tangents to the curve at these points are horizontal. The x-coordinate of R is \(a\) , and the x-coordinate of S is \(b\) , \(b \gt a\).

(h) Write down the value of \(a\) ;

(i) Write down the value of \(b\).

(j) Describe the behaviour of \(f(x)\) for \(a \lt x \lt b\).

Worked Solution

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5.	IB Standard

Let \(f(x)=jx^3+jx^2+kx+m\) where \(j, k\) and \(m\) are constants.

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

(b) If \(f'(x)\ge 0\), show that \(j^2\le 3jk\).

Worked Solution

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6.

A-Level

The following equation defines a curve which passes through \( A( 2 \pi ,3 \pi)\)

$$ \sin 2y + y \cos^22x = x + c $$

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

(b) Find an expression for \( \frac{dy}{dx} \)

(c) Find the equation of the normal to the curve at A, giving your answer in the form \(y = mx + b \)

Worked Solution

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