Exercise 1 - Exercise One 1 Show that the following equations have at least one solution in the given intervals a x cos x \u2212 2x2 3x \u2212 1 = 0

# Exercise 1 - Exercise One 1 Show that the following...

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Exercise One1. Show that the following equations have at least one solution in the given intervals.a.xcosx2x2+ 3x1 = 0 on [0.2,0.3] and [1.2,1.3]b.x(lnx)x= 0 on [4,5]2. Find intervals containing solutions to the following equationsx3+ 4.001x2+ 4.002x+ 1.101 = 0.3. Find maxaxb|f(x)|for the following functions and intervalsf(x) =2ex+2x3[0,1]4. Use the intermediate value theorem and Rolle’s theorem to show that the graphf(x) =x3+ 2x+kcrosses thex-axis exactly once, regardless of the value of the constantk.5. SupposefC[a, b] andf(x) exists on (a, b). Show that iff(x)negationslash= 0 for allxin (a, b), then therecan exist at most one numberpin [a, b] such thatf(p) = 0.6.a. Find the second Taylor polynomialP2(x) for the functionf(x) =excosxaboutx0= 0.b. UseP2(0.5) to approximatef(0.5). Find an upper bound for error|f(0.5)P2(0.5)|using the errorfunction and compare it to the actual error.c. Find a bound for the error|f(x)P2(x)|in usingP2(x) to approximatef(x) on the interval [0,1].d. Approximateintegraltext10f(x)dxusingintegraltext10P2(x)dx.e. Find an upper bound for the error in (d) usingintegraltext10R2(x)dxand compare the bound to the actualerror.7. Letf(x) = (1x)1andx0= 0. Find thenth Taylor polynomialPn(x) forf(x) aboutx0. Find avalue ofnnecessary forPn(x) to approximatef(x) to within 106on [0,0.5].8. The polynomial