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Applications of Derivatives [Video Tutorial]

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


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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