Writing a Limit as a Definite
Limit Interval
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Calculus
- Advanced Calculus: Use the Bolzano–Weierstrass Theorem to prove that if f is a continuous function on [a,b], then f is bounded on [a,b] (that is, there exists M > 0 such that |f(x)| ≤ M for all x ∈[a,b]). (Hint: Give a proof by contradiction.)arrow_forwardLet f and h be real-valued functions continuous on [a, b], differentiable on (a, b), and h(a) not equal h(b). Prove c exists in (a, b) so that (f(b)-f(a))h'c=f'(c)(h(b)-h(a))arrow_forwardDetermine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The Intermediate Value Theorem guarantees that f(a) and f(b) differ in sign when a continuous function f has at least one zero on [a, b].arrow_forward
- (Advanced Calculus) Determine whether f (t) is at least piecewise continuous in the interval [0, 10].arrow_forwarda) Give an example of a function f : [−2, 3] → R which is not continuous at 1 but which is integrable b) Give an example of a function f : [−2, 2] → R which is not differentiable at −1 but which is continuous at −1 - please include all steps and working with explanationarrow_forwardUsing the definition of monotonicity prove that the cubic function f(x)=x3 is strictly increasing for all x∈R.arrow_forward
- Functions Let the function f be differentiable on aninterval I containing c. If f has a maximum value at x = c,show that −f has a minimum value at x = c.arrow_forwardLimit and Continuity In Exercises , find the limit (if it exists) and discuss the continuity of the function. 14. lim (x, y)→(1, 1) (xy) /(x^2 − y^2 ) 16. lim (x, y)→(0, 0) (x^2 y) /(x^4 + y^2)arrow_forwardRolle's theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) such that f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b. Is it True or False?arrow_forward
- Symmetry Principle Let R be the region under the graph of y = f (x) over the interval [−a, a], where f (x) ≥ 0. Assume that R is symmetric with respect to the y-axis. (a) Explain why y = f (x) is even—that is, why f (x) = f (−x). (b) Show that y = xf (x) is an odd function. (c) Use (b) to prove that My = 0. (d) Prove that the COM of R lies on the y-axis (a similar argument applies to symmetry with respect to the x-axis).arrow_forwardThe Intermediate Value Theorem Let f be continuous over a closed, bounded interval [a,b]. If z is any real number between f(a) and f(b), then there is a number c in [a,b] satisfying f(c)=z In the following exercises, use the Intermediate Value Theorem (IVT). 151. A particle moving along a line has at each time t a position function s(t), which is continuous. Assume s (2) = 5 and s(5)=2. Another particle moves such that its position is given by h(t)=s(t)−t. Explain why there must be a value c for 2<c<5 such that h(c)=0.arrow_forwardLet f(x) = 3x + 1. a) Sketch the graph of f and shade the region below the graph of f and above the x-axis between the lines x = 0 and x = 1.b) c) d) e)Use the right endpoints of a partition {0, 14, 24,, 43,1} of [0,1], to find a Riemann sum of f over [0,1]. Usetherightendpointsofapartition{0,n1,n2,···,n =1}of[0,1],tofindaRiemannsumoffover[0,1].Use the Riemann sum in c) above to find the area of the region in a).Use the Fundamental Theorem of Calculus to find the area of the region in a).arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning