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...
Problem 1: Determine whether the statement is true or false. If the statement is true, then prove it. Otherwise, provide a counterexample. (a) If a continuous function f:R +R is bounded, then f'(2) exists for all x. (b) Suppose f.g are two functions on an interval (a, b). If both f + g and f - g are differentiable on (a, b), then both f and g are differentiable on (a,b). Problem 2: Define functions f,g: RR by: x sin(-),...
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. (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...
4. True or False. Write true or false in the blanks. a, A continuous function over a closed interval will achieve exactly one local maximum on that interval ______________ b. If f(x) and g(x) both have a local maximum at x=a then has either a local maximum or a local minimum at x=a. ___________ c. If for all x and if a > b, then _____________ d. If is undefined, and if is continuous at x=c, then has a local...
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,...
plz explain why (a) If f(x) is continuous on (a, b), then f(x) +\J(r)l do s ſ. 2. \f(x)\dx Answer: True False (b) ST sec(x) tan(x) dx = sec(x)*/3 = sec(7) - sec(1/3) = -1 -2 = -3 #/3 Answer: True False 1 = +00 (c) If an > 0 for all n > 1 and a, converges, then lim 100 Answer: True False
-/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...
Write ‘T' for true or ‘F' for false. You do not need to show any work or justify your answers for this question. The questions are 2 points each. (a) __If (xn) is a convergent sequence (converging to a finite limit) and f:RR is a continuous function, then (f (xn)) is a convergent sequence. (b) _If (xn) is a Cauchy sequence with Yn € (0,1) and f :(0,1) + R is contin- uous, then (f(xn)) is also a Cauchy sequence....
11. Circle true or false. No justification is needed. (14 points) (a) If f(x) - o(g(x), and both functions are continuous and positive, then fix dz converges. TRUE FALSE (b) If f(x)- o(g(x)), then f(x)gx)~g(x). TRUE FALSE (c) If the power series Σ an(x + 2)" converges atェ= 5, then it must km0 converge at =-6. TRUE FALSE (d) There exists a power series Σ akz" which converges to f(z)-I on some interval of positive length around FALSE TRUE (e)...
TRUE/FALSE If f(x) has a minimum at x=a, then there exists an ε, such that f(x) > f(a) for every x in (a- ε, a+ ε). If x=c is an inflection point for f, then f(c) must be a local maximum or local minimum for f. f(x) = ax2 +bx +c, (with a ≠ 0), can have only one critical point. Second Shape Theorem includes the converse of First Shape Theorem. If f’(x)=g’(x) then f(x)=g(x) +c, where c is a...