Practice Problems

Site: Saylor Academy
Course: MA005: Calculus I
Book: Practice Problems
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Date: Tuesday, July 23, 2024, 5:27 AM

Description

Work through the odd-numbered problems 1-23. Once you have completed the problem set, check your answers.

Table of contents

Problems

In problems 1–3, state each answer in the form "If \mathrm{x} is within ________ units of..."

1. \lim\limits_{x \rightarrow 3} 2 x+1=7. What values of x guarantee that f(x)=2 x+1 is (a) within 1 unit of 7?

(b) within \mathrm{0.6} units of \mathrm{7}?

(c) within \mathrm{0.4} units of \mathrm{7}?

(d) within \varepsilon units of \mathrm{7}?

3. \lim\limits_{x \rightarrow 2} 4 x-3=5. What values of x guarantee that f(x)=4 x-3 is within \mathrm{1} unit of \mathrm{5}?

(b) within \mathrm{0.4} units of \mathrm{5}?

(c) within \mathrm{0.08} units of \mathrm{5}?

(d) within \varepsilon units of \mathrm{5}?

5. For problems 1-3, list the slope of each function \mathrm{f} and the \delta (as a function of \varepsilon). For these linear functions \mathrm{f}, how is \delta related to the slope?

7. You have been asked to cut three boards (exactly the same length after the cut) and place them end to end. If the combined length must be within 0.06 inches of 30 inches, then each board must be within how many inches of 10?

9. \lim\limits_{x \rightarrow 2} x^{3}=8. What values of x guarantee that f(x)=x^{3} is within 0.5 unit of 8? within 0.05 units?

11. \lim\limits_{\mathrm{x} \rightarrow 3} \sqrt{1+x}=2. What values of \mathrm{x} guarantee that \mathrm{f}(\mathrm{x})=\sqrt{1+\mathrm{x}} is within 1 unit of 2? Within 0.0002 units?

13. You have been asked to cut four pieces of wire (exactly the same length after the cut) and form them into a square.

If the area of the square must be within 0.06 inches of 25 inches, then each piece of wire must be within how many inches of 5?

In problems 15-17, \quad \lim\limits_{\mathrm{x} \varnothing \mathrm{a}} \mathrm{f}(\mathrm{x})=\mathrm{L} and the function \mathrm{f} and \mathrm{a} value for \varepsilon are given graphically. Find a length for \delta that satisfies the definition of limit for the given function and value of \varepsilon.

15. \mathrm{f} and \varepsilon as shown in Fig. 17


17. \mathrm{f} and \varepsilon as shown in Fig. 19


In problems 19–21, use the definition to prove that the given limit does not exist.

(Find \mathrm{a} value for \varepsilon > 0 for which there is no \delta that satisfies the definition).

19. \mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}4 & \text { if } \mathrm{x} < 2 \\ 3 & \text { if } \mathrm{x} > 2\end{array} \quad\right. as is shown in Fig. 20.


Show \lim\limits_{\mathrm{x} \rightarrow 2} \mathrm{f}(\mathrm{x}) does not exist.

21. \mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}\mathrm{x} & \text { if } \mathrm{x} < 2 \\ 6-\mathrm{x} & \text { if } \mathrm{x} > 2\end{array}\right.. Show \lim\limits_{\mathrm{x} \rightarrow 2} \mathrm{f}(\mathrm{x}) does not exist.

23. Prove: If \lim\limits_{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x})=\mathrm{L} and \lim\limits_{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{g}(\mathrm{x})=\mathrm{M}, then \lim\limits_{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x})-\mathrm{g}(\mathrm{x})=\mathrm{L}-\mathrm{M}.

Answers

1. (a) If x is within \mathbf{0. 5} unit of 3.

(b) If x is within \mathbf{0. 3} unit of 3.

(c) If \mathrm{x} is within \mathbf{0. 0 2} unit of 3.

(d) If x is within \varepsilon / 2 unit of 3.

3. (a) If x is within 0.25 unit of 2.

(b) If x is within 0.1 unit of 2.

(c) If \mathrm{x} is within \mathbf{0. 0 2} unit of 2.

(d) If \mathrm{x} is within \varepsilon / 4 unit of 2.

5. Problem 1: slope =2, \delta=\varepsilon / 2. Problem 3: slope =4, \delta=\varepsilon / 4.

General pattern: \delta=\varepsilon / slope for linear functions

7. Each board must be within 0.06 / 3=0.02 inches of 10 inches in length.

(b) 1.995824623 < x < 2.004158016

9. (a) 1.957433821 < x < 2.040827551 (b) 1.995824623 < \mathrm{x} < 2.004158016

11. (a) 0 < x < 8 (b) 2.99920004 < x < 3.00080004

13. Each piece of wire must be within 0.005996404 inches of 5 inches.

15. & 17. See Figures


19. Take \varepsilon=1 / 2 (or smaller).

If x>2 and If (x)-L \mid < \varepsilon=1 / 2 then |2-L| < 1 / 2 so 3 / 2 < L < 5 / 2.

If \mathrm{x} < 2 and \mid \mathrm{f}(\mathrm{x})-L \mathrm{k} < \varepsilon=1 / 2 then \mid 3-L \mathrm{~K} < 1 / 2 so 5 / 2 < L < 7 / 2.

There is no value of \mathrm{L} that is both larger than 5 / 2 and smaller than 5 / 2 so the limit does not exist.

21. Take \varepsilon=1 / 2 (or smaller) and suppose \mathrm{x} is within 1 of 2(1 < \mathrm{x} < 3).

If 1 < \mathrm{x} < 2 and |\mathrm{f}(\mathrm{x})-L|=|\mathrm{x}-L|=|L-x| < \varepsilon=1 / 2 then -1 / 2 < L-x < 1 / 2

so x-1 / 2 < L < x+1 / 2 and L < 2.5.

If 2 < x < 3 and |f(x)-L|=|f(x)-L|=|L-6+x| < \varepsilon=1 / 2 then -1 / 2 < L-6+x < 1 / 2

so 5.5 < L+x < 7.5 and 2.5 < L.

There is no value of \mathrm{L} that is both larger than 2.5 and smaller than 2.5 so the limit does not exist.

23. This proof is very similar to the proof of the second theorem on page 9.

Assume that \lim _{x \rightarrow a} f(x)=L and \lim _{x \rightarrow a} g(x)=M. Then, given any \varepsilon>0, we know \varepsilon / 2>0 and that there are deltas for \mathrm{f} and \mathrm{g}, \delta_{\mathrm{f}} and \delta_{\mathrm{g}}, so that

if |x-a| < \delta_{f}, then |f(x)-L| < \varepsilon / 2 ("if x is within \delta_{f} of a, then f(x) is within \varepsilon / 2 of L", and

if |x-a| < \delta_{g}, then |g(x)-M| < \varepsilon / 2 ("if x is within \delta_{g} of a, then g(x) is within \varepsilon / 2 of M").

Let \delta be the smaller of \delta_{f} and \delta_{g}. If |x-a| < \delta then |f(x)-L| < \varepsilon / 2 and |g(x)-M| < \varepsilon / 2

so \mid(\mathrm{f}(\mathrm{x})-\mathrm{g}(\mathrm{x}))-(\mathrm{L}-\mathrm{M}))|=|(\mathrm{f}(\mathrm{x})-\mathrm{L})+(\mathrm{M}-\mathrm{g}(\mathrm{x})) \mid (rearranging the terms)

\leq|f(x)-L|+|M-g(x)| \quad (by the Triangle Inequality for absolute values)

 < \frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon. \quad (by the definition of the limits for \mathrm{f} and \mathrm{g}).