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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