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Express the limits in Exercises 1−8 as definite
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- 3. Consider a function f(x) sin(a²) : R → R. x²+1 (а) Show that f is continuous on R.arrow_forward3. Let f: [-2, 2] → R be defined by I f(x):= ²+1' *€ [−2,2]. Find the intervals in which f is either increasing or decreasing. Find the maxima and minima of f.arrow_forwardPlease provide hand written notesarrow_forward
- Please give me answer very fast in 5 min anuarrow_forwardWhich of the following functions f: R → R is continuous at the point 0 € R? f(x) = f(x) = f(x) = f(x) = √√√x 0 { f(x) = x² L'x X +2 {X² if x ≥ 0 if x 0 if x 0 if x 0 Choose... Choose... Choose... Choose... Choose... (▶arrow_forward1. Consider the function f defined on [0, 0), 1 x" sin=, x # 0 f(x) = x = 0 where r > 0. Determine the range of r in which (a) f is continuous on [0, c0), (b) f is differentiable on [0,0), (c) f' exists and is differentiable on [0, 0).arrow_forward
- 1. Let f: R² →→ R be defined by f(x, y) = sin(+²) and f(0,0) = 0. Is f continuous at (0, 0). Is it possible to redefine the function f at the origin in order to make the resulting new function continuous?arrow_forward7. Let (.) be the integral inner product on C[-1, 1]. That is, given f = f(x) and g = g(x), we have (f,g) = cos x are orthogonal in this f₁f(x)g(x)dx. Determine whether or not ƒ = f(x) = sin x and g = g(x) : space. Justify your answer. =arrow_forward4. Let f : R" → R be defined by f(x1, x2,... , Xn) = x1X2° ..· Xn on the cube [0, 1] × [0, 1] × ·.. x [0, 1] (i.e. for 0 < x1 < 1,0 < x2 < 1,...,0 < xn < 1). Evaluate 1 ,In) dx1 dx2 dan .. .. 1 Use your result to calculate X2, , Xn) dx1 dx2 dxn n=0 5. The average value favg of the function f : R? →R over the domain D is given 1 by the formula favg = iD // f(r, y) dA, where m(D) is the measure of the size of D (in general, this could be length, area, volume, etc.) Find the average value of the function f(x, y) = x sin (xy) on the square [0, T] × [0, 7].arrow_forward
- Let f : R → R be a twice differentiable function such that f" + f = 0. Prove there exist constants cl and c2 such that, for all real x, f(x) = c1sinx + c2cosx. We are assuming here that we know all the basic properties of sines and cosines, such as (sinx)' =cosx. (Hint: what can you say about the functions f(x)cosx – f'(x)sinx and f(x)sinx + f'(x)cos 2?)arrow_forward1. Evaluate each definite integral using the limit definition. 4r? dz -1 (b) (x² +1) d.r 1arrow_forward1. Evaluate the improper integral 2x² +3 24+1 [ 22 -dx, [Hint: consider function f(2)=23, and the contour C consisting of C₁ line segment 2 = x, x € [-R, R]; and CR the upper half circle of |z| = R]arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage