Lab 15: The fundamental theorem of calculus - part 1

  1. Part I of the Fundamental Theorem of Calculus is often illustrated beginning with the computation \( \displaystyle \frac{d}{dx} \int_a^x f(t)\ dt = \lim_{h\to 0} \frac{\int_x^{x + h} f(t)\ dt}{h} \). Draw a large graph illustrating this equality. Label everything possible, including \(t,\ a,\ x,\ h,\ x + h,\ f(t),\ \int_a^x f(t)\ dt,\) and \(\int_x^{x + h} f(t)\ dt\).


  2. Describe the meaning of the derivative in Question 1 in terms of a rate of change involving two quantities represented in your graph. Also describe what average rate is used to approximate this derivative.


  3. How could you find a good over-estimate and under-estimate for the derivative? Include representations of your over-estimate and under-estimate on your graph. Use your graph to explain why the average rate identified in Question 2 is between the over-estimate and under-estimate.


  4. Write an expression for a bound on the error. How can this bound be made small? What assumptions must be made about the function \(f\) in order to do this? Give an example of a function and value of \(x\) for which \( \displaystyle \frac{d}{dx} \int_a^x f(t)\ dt \neq f(x)\).