1. Let f: [1,4] R be defined by f(x) = 4x for 0x 2 and...

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1. Let f: [1,4] → R be defined by f(x) = 4x² for 0≤x≤ 2 and f(x) 2 < x≤ 4. Let P = {-1,0,2,3,4}. Find L(f, P) and U(f, P). = 2x for 2. Let f [0, 1] → R be defined by f(x) = -8. Prove, directly from the definition, that f is integrable. 3. Let f: [0, 1] → R with f(x) = -2 if x is rational and f(x) = 1 if z is irrational. Prove whether or not f is integrable. 4. Let f [a, b] → R be integrable. Use the definition OR the Archimedes-Riemann Theorem to prove directly that -5f is integrable and that ·b ["^(-5) = (-5) 5) [² f (Be careful, because the constant is negative the sups and infs will switch.) 1. Let f: [1,4] → R be defined by f(x) = 4x² for 0≤x≤ 2 and f(x) 2 < x≤ 4. Let P = {-1,0,2,3,4}. Find L(f, P) and U(f, P). = 2x for 2. Let f [0, 1] → R be defined by f(x) = -8. Prove, directly from the definition, that f is integrable. 3. Let f: [0, 1] → R with f(x) = -2 if x is rational and f(x) = 1 if z is irrational. Prove whether or not f is integrable. 4. Let f [a, b] → R be integrable. Use the definition OR the Archimedes-Riemann Theorem to prove directly that -5f is integrable and that ·b ["^(-5) = (-5) 5) [² f (Be careful, because the constant is negative the sups and infs will switch.)

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ANSWER 1 To find the lower sum Lf P we need to evaluate f at the left endpoint of each subinterval o... View the full answer