This tutorial covers material encountered in chapter 10 of the VCE Mathematical Methods Textbook, namely:
- Tangents and normals
- Finding and classifying stationary points
- Maximum and minimum values of a function
- Motion in 1-dimension
Q1 – Find the Tangent Equation at the Given Point
Q2 – Find the Tangent Equation at the Given Point and More
Q3 – Rate of Change of the Area of a Circle
Q4 – Stationary Points of Functions
Q5 – Minimum Value of a Function
Q6 – Motion in 1-Dimension – Velocity and Acceleration
Q7 – Rate of Increase
Worksheet
Q1. Find the equation of the tangent to the following functions at the given points
(a) \(f(x)=x^3-7x^2+5x \), at \( x=2 \)
(b) \(f(x)=x^3-7x^2+14x-3\), at [/latex]x=1[/latex]
(c) \(g(x)=\ln(x+1)\), at \(x=e-1\)
(d) \(g(x)=3\sin(\frac{x}{2})\), at \(x=\frac{\pi}{2}\)
(e) \(h(x)=2\cos(x)\), at \(x=\frac{3\pi}{2}\)
(f) \(h(x)=\ln(x^2)\), at \(x=-\sqrt{e}\)
Q2. (a) Find the equation of the tangent to the function \( f(x)=x^3-8x^2+15x \) at the point with coordinates \((4,-4)\).
(b) Find the coordinates of the point where the tangent meets \(f\) again.
Q3. Use the formula for the area of a circle \((A=\pi r^2)\) to find:
(a) The average rate of change of the area of a circle as the radius of the circle increases from \(r=3\) to \(r=4\)
(b) The instantaneous rate of change of the area with respect to the radius when \(r=4\)
Q4. Find the stationary points of the following functions and classify their nature:
(a) \(f(x)=2x^3-3x^4\)
(b) \(g(x)=x^3-4x+2\)
(c) \(h(x)=x^3-6x^2+3\)
Q5. Find the absolute minimum and its value of \(f(x)=e^{2x}+e^{-2x}\) for \(x\in[-3,3]\)
Q6. A car is travelling in a straight line away from a point \(P\). Its distance from \(P\) after \(t\) seconds is \(\frac{1}{4}e^t\) metres. Find the velocity and acceleration of the vehicle at \(t=0, 1, 2, 4\)
Q7. The diameter (\(D\) cm) of a read oak tree (not to be confused with red oak), \(t\) years after March 21, 2000 is given by \(D=40e^{kt},\,k\in\R\)
(a) Show that \(\dfrac{dD}{dt}=cD\) for some constant \(c\)
(b) If \(k=0.3\) find the rate of increase of \(D\) when \(D=120\)
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