Week 4 Exercises
4.4 Sheet 10 - Curve Sketching using differentiation
For each of the given functions find the \(y-\)intercept, \(x-\)intercepts, local maxima, local minima and points of inflection. Sketch the graph of the function clearly indicating these points.
- \(y=x^2+3x-2\).
- \(y=x^3+2x+3\).
- \(f(x)=x^4-6x^2\).
- \(f(x)=x^5-5x^4\).
- \(g(x)=\sin(x)\).
- \(g(x)=\cos(x)\).
- \(y=1/(x+1)\).
- \(y=1/(x^2+1)\).
- \(f(x)=e^x\).
- \(h(x)=\ln(x)\).
[Solutions: 1. \((0,-2), (-3.56,0), (0.56,0), (-3/2,-17,4)\) is a minimum; 2. \((0,3), (-1,0), (0,3)\) is point of inflection; 3. \((0,0), (0,0), (\pm\sqrt{6},0), (0,0)\) is a maximum, \((\pm\sqrt{3},-9)\) are minima, \((\pm 1,-5)\) are points of inflection; 4. \((0,0), (0,0), (5,0), (0,0)\) is a maximum, \((4,-256)\) is a minimum, \((0,0), (3,-162)\) are points of inflection; 5. \((0,0), (0,0), (\pi,0), (\pi/2,1)\) is a maximum, \((3\pi/2,-1)\) is a minimum; 6. \((0,1), (\pi/2,0), 3\pi/2,0), (0,1)\) is a maximum, \((\pi,-1)\) is a minimum; 7. \((0,1)\); 8. \((0,1), (0,1)\) is a maximum, \((\pm\sqrt{1/3},3/4)\) are points of inflection; 9. \((0,1)\); 10. \((1,0)\)]
4.5 Sheet 11 - Problems involving maxima & minima
Exercise 1
A beam 3 m long whose weight is 2 kN, is simply supported at both ends and the bending moment at a distance \(x\ m\) from one end is given by \[ m = 2(3x - x^2)\ kNm \] Prove that the bending moment is a maximum in the middle of the beam and calculate its value.
Exercise 2
If \(150\ m\) of fencing is to be used to enclose three sides of a rectangular plot of land, whilst the remaining side is to be a river bank, determine the lengths of the sides such that the area enclosed will be a maximum.
Exercise 3
A totally enclosed cylinder of radius \(r\), has a surface area of \(100 m^2\). Prove that the volume of the shape, \(V\), is given by \(V = 50r - \pi r^3\) and determine the value of \(r\) which maximize the volume.
4.6 Sheet 11A - Logarithmic Differentiation
Use logarithmic differentiation to find \(dy/dx\) when
- \(y = (5x+2)(3x-7)\).
- \(y = \frac{3x+5}{(x-3)(x+4)}\).
- \(y = \frac{3x+5}{x^2+x-12}\).
- \(y = \frac{x^2-12x+26}{(x-2)(x-3)(x-4)}\).
- \(y = \frac{x+3}{(x+2)^2(x-1)}\).
- \(y = \frac{7x^2+9x-1}{(3x-2)^4}\).
- \(y = \frac{(x+2)^2(2x-5)}{(6x+5)^3(x^3+4)^2}\).
- \(y = x^{\sin(x)}\).
- \(y = x^x\).
- \(y = a^x\).
[Solutions: 1. \((5x+2)(3x-7)\left(5/(5x+2)+3/(3x-7)\right)\); 2. \((3x+5)/((x-3)(x+4))\left(3/(3x+5)-1/(x-3)-1/(x+4)\right)\); 3. \((3x+5)/(x^2+x-12)\left(3/(3x+5)-(2x+1)/(x^2+x-12)\right)\); 4. \((x^2-13x+26)/((x-2)(x-3)(x-4))\left((2x-13)/(x^2-13x+26)-1/(x-2)-1/(x-3)-1/(x-4)\right)\); 5. \((x+3)/((x+2)^2(x-1))\left(1/(x+3)-2/(x+2)-1/(x-1)\right)\); 6. \((7x^2+9x-1)/(3x-2)^4\left((14x+9)/(7x^2+9x-1)-12/(3x-12)\right)\); 7. \((x+2)^2(2x-5)/((6x+5)^3(x^3+4)^2)\left(2/(x+2)+1/(2x-5)-18/(6x+5)-6x^2/(x^3+4)\right)\); 8. \(x^{\sin(x)}\left(\cos(x)\ln(x)+\sin(x)/x\right)\); 9. \(x^x(\ln(x)+1)\); 10. \(a^x\ln(a)\)]
4.7 Sheet 11B - Further Differentiation
Exercise 1
Differentiate the following implicit functions
- \(x^2+y^2-5x-6y+8=0\);
- \(3x^2+2xy+3y^2=0\);
- \(x^3+y^3+8x^2y=25\).
Exercise 2
Find \(dy/dx\) for the following functions quoted in parametric form where \(t\) and \(\theta\) are parameters and \(a\) is a constant.
- \(y=2at, x = at^2\);
- \(y=t^2, x = 2t\);
- \(y=3\cos(\theta)-\sin^3(\theta), x = \cos^3(\theta)\).
Exercise 3
Obtain \(dy/dx\) for each function by writing it in logarithmic form
- \(y=\frac{x^2\sin(x)}{\cos(2x)}\);
- \(y=x^5e^{3x}\tan(x)\);
- \(y=\frac{(3x+2)\cos(2x)}{e^{2x}}\).
[Solutions: 1(i) \((2y-6)dy/dx+2x-5=0\), (ii) \((2x+6y)dy/dx+6x+2y=0\), (iii) \((3y^2+8x^2)dy/dx+3x^2+16xy=0\); 2(i) \(dy/dx = 1/t\), (ii) \(dy/dx = t\), (iii) \(dy/dx = (1+\sin(\theta)\cos(\theta))/\cos^2(\theta)\); 3 (i) \(dy/dx=x^2\sin(x)/\cos(x)\left(2/x+\cos(x)/\sin(x)+2\sin(2x)/\cos(2x)\right)\), (ii) \(dy/dx = x^5e^{3x}\tan(x)\left(5/x+3+\sec^2(x)/\tan(x)\right)\), (iii) \(dy/dx = ((3x+2)\cos(2x))/e^{2x}\left(3/(3x+2)-2\sin(2x)/\cos(2x)-2\right)\)]