calculus Q 1, 3 Work out the following problems on a separate sheet of paper. Wr
ID: 2873332 • Letter: C
Question
calculusQ 1, 3 Work out the following problems on a separate sheet of paper. Write legibly, staple and show all your work. DO NOT turn in your rough draft. (T/F) Determine whether the following statements are true or false. If true, explain your reasoning. If false, provide a counterexample. If f is continuous at a, then lim x rightarrow a, f(x) exists. d/dx i/x^2 = 2/x^3. Evaluate lim x rightarrow 1 sin^-1(squareroot x-1/x-1). A cubic polynomial 2x^3 + 4x^2 - 8x - 5 has three roots. Using the Intermediate Value Theorem, identify three different intervals of size 1 where each of the roots can be found. (Do not use graphing utilities.) Compute the derivative d/dx(squareroot x-2/x +3e^x-pi^2+e^2).
Explanation / Answer
1. a) True. If lim x-->a f(x) exists and equal to f(a) then we say that the function f(x) is continous at x=a. So by definition we can say that if f is continous at x=a then limit exist at x-->a.
b) False. d/dx (1/x^2) = d/dx (x^(-2)) = -2/x^3 . Reason is derivative of x^n= n.x^(n-1)
3. Intermidiate value theorem states if the function is continous in the interval [a,b] then there exists a<c<b such that
f(c) lies in between f(a) and f(b).
As our function is continous from -infinity to infinity so
We need to find f(c)=0
Intervals are (1,2) , (-1,0) , (-4,-3) In all these intervals lower and upper limit function values are additive inverse of each other so there must exist f(c) = 0 by IVT.
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