Fundamental Theorem of Calculus: Accumulation and Endpoint Evaluation

All flashcards

What is a derivative?

A derivative gives the instantaneous rate of change of a function, such as the slope of its graph at a point.

What is an integral?

An integral accumulates contributions over an interval; a definite integral often represents signed area or a total amount.

What is an antiderivative?

A function $F$ is an antiderivative of $f$ when $F'(x)=f(x)$.

What does the area function $A(x)=\int_a^x f(t)\,dt$ represent?

It represents accumulated signed area under $f$ from the fixed starting point $a$ to the variable endpoint $x$.

Why is the fundamental theorem historically important?

It revealed that calculating areas and calculating slopes are linked, unifying integration and differentiation.

Quiz questions

1. If $f$ is continuous and $G(x)=\int_a^x f(t)\,dt$, what is $G'(x)$ for interior points?Answer: $f(x)$
2. For suitable functions, what is $\frac{d}{dx}\int_a^{g(x)} f(t)\,dt$?Answer: $f(g(x))g'(x)$
3. If $F'=f$ and the hypotheses of the endpoint formula hold, how is $\int_a^b f(x)\,dx$ evaluated?Answer: $F(b)-F(a)$
4. Which conditions are sufficient for the broader endpoint formula described for possibly discontinuous $f$?Answer: $f$ is Riemann integrable, and $F$ is continuous with $F'=f$ in the interior.
5. In the partition proof of the endpoint formula, what do the sums $\sum_i f(c_i)\Delta x_i$ approach as the partition is refined?Answer: The definite integral of $f$ over the interval.