A Preview of Calculus

For the following exercises (1-3), points $P(1,2)$ and $Q(x,y)$ are on the graph of the function $f(x)=x^2+1$.

1. 1. $2.2100000$
   2. $2.0201000$
   3. $2.0020010$
   4. $2.0002000$
   5. $(1.1000000, 2.2100000)$
   6. $(1.0100000, 2.0201000)$
   7. $(1.0010000, 2.0020010)$
   8. $(1.0001000, 2.0002000)$
   9. $2.1000000$
   10. $2.0100000$
   11. $2.0010000$
   12. $2.0001000$
2.  
3. $y=2x$
4.  
5. $3$
6.  
7. 1. $2.0248457$
   2. $2.0024984$
   3. $2.0002500$
   4. $2.0000250$
   5. $(4.1000000,2.0248457)$
   6. $(4.0100000,2.0024984)$
   7. $(4.0010000,2.0002500)$
   8. $(4.00010000,2.0000250)$
   9. $0.24845673$
   10. $0.24984395$
   11. $0.24998438$
   12. $0.24999844$
8.  
9. $y=\frac{x}{4}+1$
10.  
11. $\pi$
12.  
13. 1. $−0.95238095$
    2. $−0.99009901$
    3. $−0.99502488$
    4. $−0.99900100$
    5. $(−1;.0500000,−0;.95238095)$
    6. $(−1;.0100000,−0;.9909901)$
    7. $(−1;.0050000,−0;.99502488)$
    8. $(1.0010000,−0;.99900100)$
    9. $-0.95238095$
    10. $−0.99009901$
    11. $−0.99502488$
    12. $−0.99900100$
14.  
15. $y=−x-2$
16.  
17. $−49$ m/sec (velocity of the ball is $49$ m/sec downward)
18.  
19. $5.2$ m/sec
20.  
21. $-9.8$ m/sec
22.  
23. $6$ m/sec
24.  
25. Under, $1$ unit²; over: $4$ unit². The exact area of the two triangles is $\frac{1}{2}(1)(1)+\frac{1}{2}(2)(2)=2.5 \text{ units}^2$.
26.  
27. Under, $0.96$ unit²; over, $1.92$ unit². The exact area of the semicircle with radius $1$ is $\frac{\pi (1)^2}{2}=\frac{\pi }{2}$ unit².
28.  
29. Approximately $1.3333333$ unit²

Introduction to the limit of a function

In the following exercises (1-3), set up a table of values to find the indicated limit. Round to eight digits.

1. 1. $−0.80000000$
   2. $−0.98000000$
   3. $−0.99800000$
   4. $−0.99980000$
   5. $−1.2000000$
   6. $−1.0200000$
   7. $−1.0020000$
   8. $−1.0002000;$

   $\underset{x\to 1}{\lim}(1-2x)=-1$

2. 1. $−37.931934$
   2. $−3377.9264$
   3. $−333,777.93$
   4. $−33,337,778$
   5. $−29.032258$
   6. $−3289.0365$
   7. $−332,889.04$
   8. $−33,328,889$

   $\underset{x\to 0}{\lim}\frac{z-1}{z^2(z+3)}=−\infty$

3. 1. $0.13495277$
   2. $0.12594300$
   3. $0.12509381$
   4. $0.12500938$
   5. $0.11614402$
   6. $0.12406794$
   7. $0.12490631$
   8. $0.12499063$

   $\underset{x\to 2^-}{\lim}\frac{1-\frac{2}{x}}{x^2-4}=0.1250=\frac{1}{8}$

4. 1. $−10.00000$
   2. $−100.00000$
   3. $−1000.0000$
   4. $−10,000.000;$

   Guess: $\underset{\alpha \to 0^+}{\lim}\frac{1}{\alpha } \cos (\frac{\pi }{\alpha })=\infty$, Actual: DNE

   

5. False; $\underset{x\to -2^+}{\lim}f(x)=+\infty$
6. False; $\underset{x\to 6}{\lim}f(x)$ DNE since $\underset{x\to 6^-}{\lim}f(x)=2$ and $\underset{x\to 6^+}{\lim}f(x)=5$.
7.  
8. $2$
9.  
10. $1$
11. DNE
12. $0$
13.  
14. DNE
15. $2$
16.  
17. $3$
18.  
19. DNE
20.  
21. $0$
22.  

23. Answers may vary.

    

24. Answers may vary.