Constant 0.5 on the interval from 0 to 0.5. Integral is product. What is it?

Answer: ______________

Which hypothesis lets the limit pass inside the integral for an increasing nonnegative sequence?

• Decreasing to negative infinity
• Monotone increasing to the limit
• No hypotheses
• Sequence is constant zero

Non-negative functions climb steadily up to f. What does monotone convergence conclude?

• The integrals climb to the integral of the limit
• The limit function must be continuous
• Every function in the climb is bounded

Dominated convergence needs one integrable function sitting above the whole sequence.

Circle one:   True   False

Which function dominates the sliding bumps?

• The constant function 1
• The limit function 0
• None with a finite integral exists

If no integrable dominating function exists, dominated convergence does not apply and limit and integral may differ. Is this correct?

Circle one:   True   False

Which sequence gives strict Fatou inequality?

• Constants 1, 1, dots
• Zeros 0, 0, dots
• Increasing to the limit
• Bumps escaping to infinity, indicator of [n, n plus 1]

In a monotone convergence argument, where is monotonicity actually used?

• To prove each function is continuous
• To order the functions alphabetically by name
• To guarantee the integrals climb along with the functions

Integrals stay at 1 but the limit integrates to 0. Which hypothesis failed?

• No climbing and no integrable ceiling, so neither theorem applies
• The bump functions are not measurable
• The limit was taken pointwise instead of uniformly

Function raises its input to the fourth power on [0, 1]. What is its integral as a decimal?

Answer: ______________