Week 2 Exercises

2.4 Sheet 4 Tangents and Normals

Exercise 1

In each of the examples below determine the equations of the tangent and normal at the point indicated.

  1. \(y=x^2\) at the point where \(x=1\),
  2. \(y=2x^2-3x\) at the point where \(x=2\),
  3. \(y=1+x-x^2\) at the point where \(x=2\),
  4. \(y=x^3-5x+3\) at the point where \(x=3\),
  5. \(y=1/x\) at the point where \(x=2\),
  6. \(y=(x-3)^2\) at the point where \(x=5\).

Exercise 2

Determine the gradient for \(y=6x^3-9x^2-20x+6\) and show that the slope is zero when \(x=5/3\) and when \(x=-2/3\). Establish the gradient at \(x=1\) and then write the equations of the tangent and the normal to the curve at this point.

[Solutions. 1: (a) \(y=2x-1\)/\(y=-x/2+3/2\); (b) \(y=5x-8\)/\(y=-x/5+12/5\); (c) \(y=-3x+5\)/\(y=x/3-5/3\); (d) \(y=22x-51\)/\(y=-x/22+333/22\); (e) \(y=-x/4+1\)/\(y=4x-15/2\); (f) \(y=4x-16\)/\(y=-x/4+21/4\)

2: \(dy/dx = 18x^2-18x-20 = 0\) gives \(x=5/3\) and \(x=-2/3\). \(dy/dx(1) = -20\). Equation of tangent is \(y=-20x+3\), equation of normal is \(y=x/20-341/20\).]

2.5 Sheet 5 The Chain Rule

Differentiate each of the following functions using the chain rule.

  1. \(y=(2x+5)^2\)
  2. \(y=(1-5x)^4\)
  3. \(s=(3t+t)^5\)
  4. \(p=(1-2q^2)^{11}\)
  5. \(y=(1-2x)^3\)
  6. \(y=(3x^2-5x+4)^4\)
  7. \(s=(4t^3+3t-2)^2\)
  8. \(y=\frac1{\sqrt{x^2-1}}\)
  9. \(y=a\sin(a\theta)\)
  10. \(y=\sin(\omega t)\)
  11. \(y=\cos(2\pi ft+\frac{\pi}2)\)
  12. \(y=\tan(4x+3)\)
  13. \(y=6e^{5x}\)
  14. \(y=\ln(2x^5)\)
  15. \(y=\ln(\frac 1x)\)

{Solutions. (1) \(dy/dx=8x+20\); (2) \(dy/dx=-20(1-5x)^3\); (3) \(ds/dt=15(3t+7)^4\); (4) \(dp/dq=-44q(1-2q^2)^{10}\); (5) \(dy/dx=-6(1-2x)^2\); (6) \(dy/dx=(24x-20(3x^2-5x+4)^3\); (7) \(ds/dt=(24t^2+6)(4t^3+3t-2)\); (8) \(dy/dx=-x/(x^2-1)^{3/2}\); (9) \(dy/d\theta=a^2\cos(a\theta)\); (10) \(dy/d\omega=\omega\cos(\omega t)\); (11) \(dy/dt=-2\pi f\sin(2\pi ft+\pi/2)\); (12) \(dy/dx=4\sec^2(4x+3)\); (13) \(dy/dx=30e^{5x}\); (14) \(dy/dx=5/x\); (15) \(dy/dx=-1/x\)}

2.6 Sheet 6 The Product Rule

\[ \text{if }y(x)=u(x)v(x)\text{ then }dy/dx=udv/dx+vdu/dx \] Differentiate the following functions

  1. \(y=(4x+5)(x^2-2x)\)
  2. \(y=x^{5/3}(4x-2)^2\)
  3. \(y=(x+1)^2(1-2x)\)
  4. \(y=4x\sqrt{5-2x^2}\)
  5. \(y=(4x^2+3x)(2x-8)^5\)
  6. \(y=8x^3\sin(3x)\)
  7. \(y=4x^2\cos(x)\)
  8. \(y=(x^2-2x)\sin(x)\)
  9. \(y=e^{2x}\sin(x)\)
  10. \(y=x\ln(x)\)
  11. \(y=\sqrt{x^2+4}\cos(2x+1)\)
  12. \(y=\sqrt{x^2-1}\ln(2x+1)\)
  13. \(y=e^{x\ln(x)}\)
  14. \(y=x^x\)

{Solutions. 1. \((4x+5)(2x-2)+4(x^2-2x)\); 2. \(8x^{5/3}(4x-2)+5x^{2/3}(4x-2)^2/3\); 3. \(-2(x+1)^2+2(1-2x)(x+1)\); 4. \(-8x^2(5-2x^2)^{-1/2}+4(5-2x^2)^{1/2}\); 5. \(10(4x^2+3x)(2x-8)^4+(8x+3)(2x-8)^5\); 6. \(24x^2\sin(3x)+24x^3\cos(3x)\); 7. \(8x\cos x-4x^2\sin(x)\); 8. \((2x-2)\sin(x)+(x^2-2x)\cos(x)\); 9. \(2e^{2x}\sin(x)+e^{2x}\cos(x)\); 10. \(\ln(x)+1\); 11. \(x(x^2+4)^{-1/2}\cos(2x+1)-2(x^2+4)^{1/2}\sin(2x+1)\); 12. \(x(x^2-1)^{-1/2}\ln(2x+1)+2\sqrt{x^2-1}/(2x+1)\); 13. \(\exp(x\ln x)(1+\ln x)\); 14. same as 13.}

2.7 Sheet 7 The Quotient Rule

Differentiate the following functions with respect to \(x\) using the quotient rule

  1. \(y=\frac{3x^2+2}{5x+4}\)
  2. \(y=\frac{(9x^2+3)^2}{2x-8}\)
  3. \(y=\frac{1+x}{1-x}\)
  4. \(y=\frac{2x+2}{(3x^3+2x)^2}\)
  5. \(y=9x^2-\frac{x^2+2x}{1-x}\)
  6. \(y=\frac{3x-4}{2x^2+1}\)
  7. \(y=\frac{\sin(3x)}{x+1}\)
  8. \(y=\frac{4x(3x-4)^2}{\sin(x)}\)

{Solutions: 1. \(dy/dx = (15x^2+24x-10)/(5x+4)^2\); 2. \(dy/dx = 18(3x^2+1)(9x^2-48-1)/(2x-8)^2\); 3. \(dy/dx = 2/(1-x)^2\); 4. \(dy/dx = -2(3x^3+2x)(15x^3+18x^2+2x+4)/(3x^3+2x)^4\); 5. \(dy/dx = 18x+(x^2-2x-2)/(1-x)^2\); 6. \(dy/dx = (-6x^2+16x+3)/(2x^2+1)^2\); 7. \(dy/dx = (3\cos(3x)(x+1)-\sin(3x))/(x+1)^2\); 8. \(dy/dx = 4(3x-4)((9x-4)\sin(x)-x(3x-4)\cos(x))/\sin^2(x)\)}