Calculus I, Fall 2021

Brief solutions to Quiz 1

Oct 12, 2021

1. (36 pts)

(a) State precise definition of lim_{x→c^{+}}f(x) = L.

(b) Use it to prove that lim_{x→0^{+}}\sqrt{x} = 0. Ans:

Figure 1: Answer to problem 1

2. (32 pts) Evaluate lim_{θ→0}f(θ) where f(θ) = cos (\frac{sin(1 − cos θ)}{tan^{2} θ}). Take for granted that cos(·) is a continuous function. State clearly any Theorem you use, but need not prove it. Note that f(θ) is NOT defined on θ = 0. Remark: See also the hint in problem 3, homework week 04. Ans:

Figure 2: Answer to problem 2

3. (32 pts) (a) State the Intermediate Value Theorem.

(b) Use it to locate a root of x − 1 = cos x on an interval of length 1. That is, find a c such that there is a root on (c, c + 1).

Ans:

Figure 3: Answer to problem 3 3 Figure 4: Some common mistakes to problem 3

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