Calculus 1 Labs
Lab 15: The fundamental theorem of calculus  part 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 \(\int_x^{x + h} f(t)\ dt\).

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.

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

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 \( \displaystyle \frac{d}{dx} \int_a^x f(t)\ dt \neq f(x)\).