MA005: Calculus I

1. $2^{2}+3^{2}+4^{2}=29$

3. $(1+1)^{2}+(1+2)^{2}+(1+3)^{2}=29$

5. $\cos (0)+\cos (\pi)+\cos (2 \pi)+\cos (3 \pi)+\cos (4 \pi)+\cos (5 \pi)=1+(-1)+1+(-1)+1+(-1)=0$

7. $\sum_{\mathrm{k}=3}^{94} \mathrm{k}$

9. $\sum_{k=3}^{12} k^{2}$

11. $\sum_{k=1}^{7} k \cdot 2^{k}$

13. (a) $(1+2)+(2+2)+(3+2)=3+4+5=12$           (b) $(1+2+3)+(2+2+2)=12$

15. (a) $5 \cdot 1+5 \cdot 2+5 \cdot 3=5+10+15=30$                       (b) $5 \cdot(1+2+3)=5 \cdot 6=30$

17. (a) $1^{2}+2^{2}+3^{2}=1+4+9=14$                                     (b) $(1+2+3)^{2}=6^{2}=36$

19. $f(0)+f(1)+f(2)+f(3)=0^{2}+1^{2}+2^{2}+3^{2}=14$

21. $2 \cdot f(0)+2 \cdot f(1)+2 \cdot f(2)+2 \cdot f(3)=2 \cdot 0+2 \cdot 1+2 \cdot 4+2 \cdot 9=28$

23. $g(1)+g(2)+g(3)=3+6+9=18$

25. $g^{2}(1)+g^{2}(2)+g^{2}(3)=3^{2}+6^{2}+9^{2}=126$

27. $\mathrm{h}(2)+\mathrm{h}(3)+\mathrm{h}(4)=\frac{2}{2}+\frac{2}{3}+\frac{2}{4}=\frac{13}{6}$

29. $\mathrm{f}(1) \mathrm{h}(1)+\mathrm{f}(2) \mathrm{h}(2)+\mathrm{f}(3) \mathrm{h}(3)=(1)(2)+(4)(1)+(9)(2 / 3)=12$

31. $\left(1^{2}-0^{2}\right)+\left(2^{2}-1^{2}\right)+\left(3^{2}-2^{2}\right)+\left(4^{2}-3^{2}\right)+\ldots+\left(7^{2}-6^{2}\right)=7^{2}-0^{2}=49$

33. $\left(\frac{1}{1}-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)+\left(\frac{1}{5}-\frac{1}{6}\right)=1-\frac{1}{6}=\frac{5}{6}$

35. $(\sqrt{1}-\sqrt{0})+(\sqrt{2}-\sqrt{1})+(\sqrt{3}-\sqrt{2})+\ldots+(\sqrt{9}-\sqrt{8})=\sqrt{9}-\sqrt{0}=3$

37. (i) $[2,3],[3,4.5],[4.5,6],[6,7]$                  (ii) $1,1.5,1.5,1$              (iii) $\text {mesh }=1.5$             (iv) $1+1.5+1.5+1=5$

39. (i) $[-3,-1],[-1,0],[0,1.5],[1.5,2]$         (ii) $2,1,1.5,0.5$             (iii) $\text {mesh }=2$                 (iv) $2+1+1.5+0.5=5$

41. (i) $[3,3.8],[3.8,4.5],[4.5,5.2],[5.2,7]$      (ii) $0.8,0.7,0.7,1.8$       (iii) $\text { mesh }=1.8$             (iv) $0.8+0.7+0.7+1.8=4$

43. $\begin{aligned} &\Delta x_{1}+\Delta x_{2}+\Delta x_{3}+\ldots+\Delta x_{n}=\left(x_{1}-x_{0}\right)+\left(x_{2}-x_{1}\right)+\left(x_{3}-x_{2}\right)+\ldots+\left(x_{n}-\right. \\ &\left.x_{n-1}\right)=x_{n}-x_{0} \end{aligned}$

45.
(a) $f(0)(1)+f(1)(0.5)+f(2)(0.5)=(4)(1)+(3)(0.5)+(0)(0.5)=5.5$
(b) $\mathrm{f}(1)(1)+\mathrm{f}(1.5)(0.5)+\mathrm{f}(1.5)(0.5)=(3)(1)+(1.75)(0.5)+(1.75)(0.5)=4.75$

47. a. (i) and (ii) See the graph
(iii) $\mathrm{f}(0)=0, \mathrm{f}(\pi / 4) \approx 0.707, \mathrm{f}(\pi / 2)=1$
(iv) $(\pi / 4)(0)+(\pi / 4)(0.707)+(\pi / 2)(1) \approx 2.13$

49.

(a) $(2)(1)+(5)(2)+(17)(1) \leq \mathrm{RS} \leq(5)(1)+(17)(2)+(26)(1) \text { so } 29 \leq \mathrm{RS} \leq 65$
(b) $(2)(1)+(5)(1)+(10)(1)+(17)(1) \leq \mathrm{RS} \leq(5)(1)+(10)(1)+(17)(1)+(26)(1) \text { so } 34 \leq \mathrm{RS} \leq 58$
(c) $(2)(0.5)+(3.25)(0.5)+(5)(1)+(10)(1)+(17)(1) \leq \mathrm{RS} \leq(3.25)(0.5)+(5)(0.5)+(10)(1)+(17)(1)+(26)(1)$ so $34.625 \leq \mathrm{RS} \leq 57.125$

51.
(a) $(0)(\pi / 2)+(0)(\pi / 2) \leq \mathrm{RS} \leq(1)(\pi / 2)+(1)(\pi / 2) \text { so } 0 \leq \mathrm{RS} \leq \pi$
(b) $(0)(\pi / 4)+(0.707)(\pi / 4)+(0)(\pi / 2) \leq \mathrm{RS} \leq(0.707)(\pi / 4)+(1)(\pi / 4)+(1)(\pi / 2) \text { so } 0.56 \leq \mathrm{RS} \leq 2.91$
(c) $(0)(\pi / 4)+(0.707)(\pi /)+(0.707)(\pi / 4)+(0)(\pi / 4) \leq \mathrm{RS} \leq(0.707)(\pi / 4)+(1)(\pi / 4)+(1)(\pi / 4)+(0.707)(\pi / 4)$ so $1.11 \leq \mathrm{RS} \leq 2.68$

53. (a) $17.402-7.362 \mid=0.04$                      (b) $|7.390-7.372|=0.018$

55. $\text { I error } \mathrm{I} \leq\left(\text { base) }(\text { height })=\frac{4-2}{50}(65-9)=\frac{2}{50}(56)=\frac{112}{50}=2.24\right.$

57. (a) $\frac{100(101)}{2}=5050$                                       (b) $\begin{aligned} &1+2+3+\ldots+100 \\ &100+99+98+\ldots+1 \\ \hline &101+101+101+\ldots+101=100(101)=10100 . \quad \frac{1}{2}(10100)=5050 \end{aligned}$    

59. $10+11+12+\ldots+20=(1+2+3+\ldots+20)-(1+2+3+\ldots+9)=\frac{20(21)}{2}-\frac{9(10)}{2}=210-45=165$

61. $\sum_{k=1}^{10}\left(k^{3}+k\right)=\sum_{k=1}^{10} \mathrm{k}^{3}+\sum_{k=1}^{10} \mathrm{k}=\left(\frac{10(11)}{2}\right)^{2}+\frac{10(11)}{2}=(55)^{2}+55=3080$