Final Exam

Question 1

If f is continuous on [a,b], state the (Riemann Sum) definition of integral of (a,b) f(x) dx

Answer 1

If f is continous on [a.b], the indefinitie integral of f is,

integral from a to b f(x) dx = limit as max Δxi approaches 0 sum of n (top) i=1 f(xi)Δ xi

provided this limit exists. If it does, we say f is integratabtle on [a,b]

Question 2

State the Fundamental Theorem of Calculus

Answer 2

Part 1: Let f(t) be continuous on [a,b]. Let g(x) = the integration of a to x f(t) dt.
Then, g(x) is continuous and differentiable and g’(x) = f(x)

Part 2: Let f be continuous on [a,b] and let F be any antiderivative of f.
Then, the integration of a to b f(x) dx =
F(x) |a to b = F(b) - F(a)

Question 3

State the definition of the derivative

Answer 3

The derivative of a function f at a number a is

f’(x) = lim h–> 0 of f(x+h)-f(x)/h provided that the limit exists