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Writing a Limit as a Definite
Limit Interval
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Calculus
- Definition: The AREA A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles.arrow_forwardQuestion: what is the notation for’ the definite integral of a continuous function f over the interval [a,b]”?arrow_forwardUsing the definition of monotonicity prove that the cubic function f(x)=x3 is strictly increasing for all x∈R.arrow_forward
- a) 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_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_forwardFunctions 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_forward
- Expand/Write out (but do NOT calculate) the Riemann sum, Rs, for the function f(x) = 1 - |x| on the interval [-1,14]. Use the function explicitly.arrow_forwardSymmetry 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_forwardAnalysis problem Prove that f(x) = x ⋅ |x| is continuous at all points c in ℝ.arrow_forward
- (Term-by-term Differentiability Theorem). Let fn be differentiable functions defined on an interval A, and assume ∞ n=1 fn(x) converges uniformly to a limit g(x) on A. If there exists a point x0 ∈ [a, b] where ∞ n=1 fn(x0) converges, then the series ∞ n=1 fn(x) converges uniformly to a differentiable function f(x) satisfying f(x) = g(x) on A. In other words, Proof. Apply the stronger form of the Differentiable Limit Theorem (Theorem6.3.3) to the partial sums sk = f1 + f2 + · · · + fk. Observe that Theorem 5.2.4 implies that sk = f1 + f2 + · · · + fk . In the vocabulary of infinite series, the Cauchy Criterion takes the followingform.arrow_forwardf(x) =√(x3+ 8) a) Use properties of the integral and show that 3 ≤ 1∫2 f(x)dx ≤ 4 b) Express the integral 1∫2 f(x)dx as a limit of a Riemann sum. Do not evaluate the limit.arrow_forwardlim ilm f(x,y)=x²y²÷x²y² Calculate the two limits of the functionarrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning