Problem 1: Determine whether the statement is true or false. If the statement is true, then...
With justification in each one. Clarification; why if true and why if false? Please Determine whether the following statement is true or false: • Iff: R+R is differentiable and strictly increasing on R, then f'(1) > 0 VI ER • If S: R R is continuous and f(x) - ron Q, then (V3) - 3. • If f,g: (0,1) - Rare functions such that \S(1)-f(y) = g(1)-9(y) for all 1, y € (0, 1) and g is continuous on (0,1),...
1. Let f:R → R be the function defined as: 32 0 if x is rational if x is irrational Prove that lim -70 f(x) = 0. Prove that limc f(x) does not exist for every real number c + 0. 2. Let f:R + R be a continuous function such that f(0) = 0 and f(2) = 0. Prove that there exists a real number c such that f(c+1) = f(c). 3 Let f. (a,b) R be a function...
7. (16 marks) True-false questions: in each case decide if the statement is true or false; if true, present a short argument supporting it, and if it is false present a counterexample. a) The product of a rational and an irrational number is always irrational. b) If x and y are non-constructible then so is x + y. c) If x is constructible then so is 1. d) The set of non-constructibles is a subfield of R. e) The set...
Determine whether the statement is TRUE or FALSE. You are NOT required to justify your answers. (a) Suppose both f and g are continuous on (a, b) with f > 9. If Sf()dx = Sº g(x)dx, then f(x) = g(x) for all 3 € [a, b]. (b) If f is an infinitely differentiable function on R with f(n)(0) = 0 for all n = 0,1,2,..., then f(x) = 0 for all I ER. (c) f is improperly integrable on (a,...
Let x,y ∈ R. Which of the following statements are true. If the statement is true prove it, if not give a counterexample a) If x is rational and y is irrational, then x y is irrational. b) If x and y are both irrational then x + y is irrational. c) Ifx and y are both irrational then ry is irrational d) Ifx is rational and y is irrational then ry is irrational.
1. Determine whether the statement is true or false. If false, explain why and correct the statement (T/FIf)exists, then lim ()f) o( T / F ) If f is continuous, then lim f(x) = f(r) (TFo)-L, then lim f(x)- lim F(x) "( T / F ) If lim -f(x)s lim. f(x) L, then lim f(x)s 1. "(T/F) lim. In x -oo . (T/F) lim0 ·(T / F ) The derivative f' (a) is the instantaneous rate of change of y...
ntifiers , Counterexamples, Disproof (#9, 15 pts) #9. For each statement, state whether the statement is true or false. If false, explain; provide a counterexample as appropriate or a careful explanation. (If true, no explanation expected) (a) n in N, n+23 ≥n3+8. (b) x in R, x+23 ≥x3+8. (c) n in N, 4n + 1 is prime. (d) x, y in R, if |x| < |y|, then x2 < xy. (e) m in N such that n in N, m...
2. Let f:R + R and g: R + R be functions both continuous at a point ceR. (a) Using the e-8 definition of continuity, prove that the function f g defined by (f.g)(x) = f(x) g(x) is continuous at c. (b) Using the characterization of continuity by sequences and related theorems, prove that the function fºg defined by (f.g)(x) = f(x) · g(x) is continuous at c. (Hint for (a): try to use the same trick we used to...
1. (3 points each) Answer each of the following statements as true or false a. If lim ) exists, then lim(lim() b. If lim f (x) exists, then fi (zo) exists. c. If f differentiable on la, b, then f is integrable on [a, b]. d. If f is continuous on [a, b] and differentiable on (a, b), then there exists a number X -To (a, b) such that f (b) f(a)- (b-a)f (x). e. If f is integrable on...
-/10 POINTS SCALCET8 5.TF.006. Determine whether the statement is true or false. 18 If f'is continuous on [7,8], then I f'(v) dv = f(8) - f(7). C True C False Need Help? Talk to a Tutor | -110 POINTS SCALCET8 5.TF.008. Determine whether the statement is true or false. If f and g are differentiable and f(x) = g(x) for a < x < b, then f '(x) > g'(x) for a < x <b. True False Need Help? Talk...