Section 7 - Applications of differentiation
The topic of applications of differentiation has been broken down into the following sections:
Exam 2: Applications of differentiation (VCAA, 2006S)
(a)
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Consider the function \(f:[0,2]\rightarrow R, f(x)=ax^3+bx^2+cx+2\).
The graph of \(y=f(x)\) has a turning point with coordinates \((1,1)\). |
i.
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Find the values of \(a\) and \(b\) in terms of \(c\).
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ii.
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If also \(f(2)=0\), find the value of \(c\).
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(b)
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Consider the function \(f:[0,2]\rightarrow R, f(x)=(x-1)^2(x-2)+1\).
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i.
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Find \(f'(x)\).
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ii.
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The coordinates of the turning point of the graphs of \(y=f(x)\) occur at \((m,1)\) and \((n,\frac{23}{27})\). Find the values of \(m\) and \(n\).
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iii.
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State the absolute maximum and minimum values of this function.
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iv.
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Find the real values of \(p\) for which the equation \(f(x)=p\), where \(x\in [0,2]\), has exactly one solution.
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(c)
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Consider the function \(f:R\rightarrow R, f(x)=(x-1)^2(x-2)+1\).
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i.
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Describe a sequence of transformations which transforms the graph of \(y=f(x)\) into the graph of \(y=f(\frac{x}{k})-1\).
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ii.
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Find the \(x\)-axis intercepts of the graph of \(y=f(\frac{x}{k})-1\).
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iii.
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Find the real values of \(h\) for which only one of the solutions of \(f(x+h)=1\) is positive.
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(d)
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The graph of \(y=(x-1)^2(x-a)\) where \(a>1\) is shown below.
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Find the exact value of \(a\) such that the local minimum at point \(A\) lies on the line with equation \(y=-4x\).
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Success Criteria
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Topic Resources
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Assessment Tasks
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VCAA Questions
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Other Resources
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Applications of differentiation:
- To be able to calculate the equation of a tangent line to a curve.
- To be able to calculate the equation of a perpendicular line to a tangent line (normal lines).
- To be able to calculate the average rate of change between two points.
- To be able to calculate the instantaneous rate of change using differentiation.
- To be able to determine the coordinates of stationary points for a graph.
- To be able to determine the nature of a stationary point.
- To be able to identify when a functions is strictly increasing or strictly decreasing over an interval.
- To be able to determine the absolute (or global) maximum or minimum of a function over a given interval.
- To understand the relationships between displacement, velocity and acceleration (Kinematics).
- To be able to use calculus to find the displacement, velocity and/or acceleration of an object.
- To be able to express a variable in terms of another variable.
- To be able to differentiate and determine any stationary points of a function.
- To be able to justify that a stationary point is a maximum or minimum point.
- To be able to use calculus to optimise the relationship between two variables.
Applications of differentiation - Worksheet A
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Applications of differentiation - Worksheet B
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Applications of differentiation - Worksheet C
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Applications of differentiation - Worksheet D
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Optimisation - Worksheet A
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Optimisation - Worksheet B
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Optimisation - Worksheet C
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Applications of differentiation - Pretest A:
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