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    "## Exercise 1 - integration of infinite intervals\n",
    "\n",
    "Calculate the value of the following integral using Gaussian quadrature: $$I=\\int_0^\\infty e^{-t^2}dt$$ The value of this integral is known analytically: $I=\\frac{1}{2}\\sqrt{\\pi}=0.886226925453\\ldots$.  \n",
    "\n",
    "To solve the integral numerically, make the change of variables $z=t/(1+t)$. The integral becomes $$I=\\int_0^1 \\frac{e^{-z^2/(1-z)^2}}{(1-z)^2}dz .$$ \n",
    "\n",
    "1. Plot the functions $ f(x)=e^{-t^2} $ and $i(z)=\\frac{e^{-z^2/(1-z)^2}}{(1-z)^2}$.\n",
    "2. Change your Gauss-Legendre integration program from last week to solve this integral. \n",
    "3. Plot the value of the integral and/or the integration error as a function of integration points.\n",
    "\n",
    "\n",
    "## Talking points\n",
    "\n",
    "1. What do you observe?\n",
    "2. How many points do you need for 0.1% accuracy and how many for $10^{-6}$% accuracy?"
   ]
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    "## Exercise 2 - multi dimensional integrals\n",
    "\n",
    "Calculate the double integral $\\DeclareMathOperator\\erf{erf} \\int_{-1}^1\\int_{-1}^1 e^{(-x^2-y^2)} dx dy = \\pi \\erf(1)^2$ numerically. \n",
    "\n",
    "1. Plot the functions $ e^{(-x^2-y^2)}$.\n",
    "2. Adapt your Gauss-Legendre integration program to solve this integral. \n",
    "3. Plot the value of the integral and/or the integration error as a function of integration points.\n",
    "\n",
    "## Talking points\n",
    "\n",
    "1. What do you observe?\n",
    "2. How many points do you need for 0.1% accuracy and how many for $10^{-6}$% accuracy?"
   ]
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   "source": [
    "## Exercise 3 - numeric differentiation\n",
    "\n",
    "Calculate the derivative of $f(x)=x^3-2x^2+x-1$ numerically with the forward $\\left( \\frac{df}{dx} \\approx \\frac{f(x+h)-f(x)}{h}\\right)$ or the backward difference $\\left( \\frac{df}{dx} \\approx \\frac{f(x)-f(x-h)}{h} \\right)$ method.  \n",
    "\n",
    "1. Use the forward and backward difference rule to calculate $f'(x)$ for $h=0.1$ and $h=0.01$.\n",
    "2. Plot the analytic derivative and your numeric derivatives in the same graph. \n",
    "3. Plot the difference between the analytic and the numeric derivatives.\n",
    "\n",
    "## Talking points\n",
    "\n",
    "1. What do you observe?\n",
    "2. How does the error in the numeric derivatives behave with $x$ and $h$?\n",
    "3. How can we do better?"
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    "## Exercise 4 - central difference method\n",
    "\n",
    "Calculate the derivative of $f(x)=x^3-2x^2+x-1$ numerically with the central difference $\\left( \\frac{df}{dx}\\approx\\frac{f((x+h/2)-f(x-h/2)}{h}\\right) $ method.  \n",
    "\n",
    "1. Use the central difference rule to calculate $f'(x)$ for $h=0.1$ and $h=0.01$.\n",
    "2. Plot the analytic and your numeric derivative in the same graph. \n",
    "3. Plot the difference between the analytic and the numeric derivative.\n",
    "\n",
    "## Talking points\n",
    "\n",
    "1. What do you observe?\n",
    "2. How does the error in the numeric derivatives now behave with $x$ and $h$?"
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