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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
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.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 \(x=-3\).
3. | 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 \(a \lt x \lt b\).
4. | IB Standard |
The following diagram shows part of the graph of \(y=f (x)\)
The graph has a local maximum where \(x=- \frac23\), and a local minimum where \(x=4\).
sketch the graph of \(y=f'(x)\) for \(-4\le x \le 7\)
Write down the following in order from least to greatest: \(f(2),f'(4)\) and \(f''(4)\).
5. | IB Studies |
Consider the function \(f(x)=\frac{20}{x^2}+kx\) where \(k\) is a constant and \(x\neq0\).
(a) Write down \(f'(x)\)
The graph of \(y = f(x)\) has a local minimum point at \(x=2\).
(b) Show that \(k=5\).
(c) Find \(f(1)\).
(d) Find \(f'(1)\).
(e) Find the equation of the normal to the graph of \(y=f(x)\) at the point where \(x=1\)
Give your answer in the form \(ay+bx+c=0\) where \(a, b, c \in \mathbb{Z}\)
(f) Sketch the graph of \(y=f(x)\) , for \(-5\le x\le 10\) and \(-10\le y\le 50\).
(g) Write down the coordinates of the point where the graph of \(y=f(x)\) intersects the x-axis.
(h) State the values of \(x\) for which \(f(x)\) is decreasing.
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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.
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