Let's solve **Question 6** step by step: --- ### **Question 6** Use a) the left end, b) the right end, and c) the trapezoidal rules to approximate the definite integral \[ \int_{-1}^{1}(x^2+1)\,dx \] using a partition of \([-1, 1]\) into 4 equal parts. --- Let's break it down: #### **Step 1: Find the partition points** The interval \([-1, 1]\) is of length \(2\). Dividing into 4 equal parts: \[ \Delta x = \frac{1 - (-1)}{4} = \frac{2}{4} = 0.5 \] So, the partition points are: \[ x_0 = -1,\quad x_1 = -0.5,\quad x_2 = 0,\quad x_3 = 0.5,\quad x_4 = 1 \] --- #### **Step 2: Calculate the function values** \(f(x) = x^2 + 1\) \[ f(x_0) = f(-1) = (-1)^2 + 1 = 2 \] \[ f(x_1) = f(-0.5) = (-0.5)^2 + 1 = 0.25 + 1 = 1.25 \] \[ f(x_2) = f(0) = 0^2 + 1 = 1 \] \[ f(x_3) = f(0.5) = (0.5)^2 + 1 = 0.25 + 1 = 1.25 \] \[ f(x_4) = f(1) = (1)^2 + 1 = 2 \] --- #### **Part a) Left endpoint rule** Left endpoint: use \(x_0, x_1, x_2, x_3\): \[ L = \Delta x [f(x_0) + f(x_1) + f(x_2) + f(x_3)] = 0.5 [2 + 1.25 + 1 + 1.25] = 0.5 \times 5.5 = 2.75 \] --- #### **Part b) Right endpoint rule** Right endpoint: use \(x_1, x_2, x_3, x_4\): \[ R = \Delta x [f(x_1) + f(x_2) + f(x_3) + f(x_4)] = 0.5 [1.25 + 1 + 1.25 + 2] = 0.5 \times 5.5 = 2.75 \] --- #### **Part c) Trapezoidal rule** Formula: \[ T = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)] \] Plug in the values: \[ T = \frac{0.5}{2} [2 + 2(1.25) + 2(1) + 2(1.25) + 2] \] \[ = 0.25 [2 + 2.5 + 2 + 2.5 + 2] \] \[ = 0.25 [11] \] \[ = 2.75 \] --- ### **Final Answers** - **Left endpoint rule:** \(2.75\) - **Right endpoint rule:** \(2.75\) - **Trapezoidal rule:** \(2.75\) --- ### **(Bonus) Exact Value** Let's quickly check the exact value of the integral: \[ \int_{-1}^1 (x^2 + 1)\,dx = \int_{-1}^1 x^2\,dx + \int_{-1}^1 1\,dx = [\frac{x^3}{3}]_{-1}^1 + [x]_{-1}^1 = \left(\frac{1}{3} - \frac{(-1)}{3}\right) + (1 - (-1)) = 0 + 2 = 2 \] The approximate rules overestimate the actual value in this case. --- **Let me know if you want to solve the next question!**

Let's proceed to **Question 7** step by step: --- ### **Question 7** #### **(a) Write \(21 = 545\) in polar form.** It seems there might be a typo or formatting issue in the question, but likely, it refers to expressing **complex numbers** in polar form. Assuming the complex numbers are: \[ z_1 = 21 \quad \text{and} \quad z_2 = 545 \] But these are real numbers, so their polar forms are straightforward: - For **real positive numbers** \(a > 0\), the polar form is: \[ a = r e^{i \theta} \] where \(r = |a|\), and \(\theta = 0\). So: \[ z_1 = 21 = 21 e^{i \cdot 0} \] \[ z_2 = 545 = 545 e^{i \cdot 0} \] --- ### **(b) Find the exact value of \(57^\circ \times 25^\circ\) in Cartesian coordinates.** Again, the notation seems ambiguous, but probably the question involves multiplying two complex numbers in polar form: Suppose: \[ z_1 = r_1 e^{i \theta_1} \quad \text{and} \quad z_2 = r_2 e^{i \theta_2} \] and the question asks for the product \(z_1 \times z_2\). Given: - \(z_1\) has modulus 57 and argument 25°. - \(z_2\) (possibly) has modulus 57 and argument 25°, or perhaps they are 57° and 25°, respectively. Assuming the latter, i.e., \[ z_1 = r_1 e^{i \theta_1} = 57 e^{i 25^\circ} \] and \[ z_2 = r_2 e^{i \theta_2} = 25 e^{i \text{(unknown)}} \] But the question seems to refer to: > Find the exact value of \(57^\circ \times 25^\circ\) which is probably about multiplying two complex numbers: \[ z_1 = 57 e^{i 25^\circ} \] \[ z_2 = 25 e^{i 57^\circ} \] then: \[ z = z_1 \times z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)} = (57)(25) e^{i(25^\circ + 57^\circ)} \] Calculate: \[ r = 57 \times 25 = 1425 \] \[ \theta = 25^\circ + 57^\circ = 82^\circ \] Now, **convert to Cartesian coordinates**: \[ z = r (\cos \theta + i \sin \theta) = 1425 (\cos 82^\circ + i \sin 82^\circ) \] Using approximate values: \[ \cos 82^\circ \approx 0.13917 \] \[ \sin 82^\circ \approx 0.99026 \] So, \[ z \approx 1425 \times 0.13917 + i \times 1425 \times 0.99026 \] \[ \approx 198.4 + i \times 1412.8 \] --- ### **Final Answer (in Cartesian form):** \[ \boxed{ z \approx 198.4 + 1412.8 i } \] --- ### **(c) Write \(z = x + i y\)** if The question snippet seems incomplete, but if you're asked to express a complex number in the form \(x + iy\), then: - \(x = \text{Re}(z)\) - \(y = \text{Im}(z)\) from the calculation above, the approximate Cartesian form is: \[ z \approx 198.4 + 1412.8 i \] --- **Let me know if you'd like me to clarify any step or proceed with the next question!**

Certainly! Let's move on to **Question 8** step by step. --- ### **Question 8** #### **(a) Use the integral test to determine whether the series \(\sum_{n=1}^\infty \frac{1}{n^p}\) converges.** *Note:* The series \(\sum_{n=1}^\infty \frac{1}{n^p}\) is known as the p-series. **Step 1: Recall the integral test** The integral test states that for a positive, decreasing function \(f(n)\), the series \(\sum_{n=1}^\infty f(n)\) converges if and only if the improper integral: \[ \int_1^\infty f(x) \, dx \] converges. **Step 2: Apply the test** Set \(f(x) = \frac{1}{x^p}\). Compute: \[ I = \int_1^\infty \frac{1}{x^p} dx \] - For \(p \neq 1\): \[ I = \left[ \frac{x^{1 - p}}{1 - p} \right]_1^{\infty} \] - For \(p > 1\): \[ \lim_{t \to \infty} \frac{t^{1 - p}}{1 - p} = 0 \quad (\text{since } p > 1 \Rightarrow 1 - p < 0) \] and at \(x=1\): \[ \frac{1^{1 - p}}{1 - p} = \frac{1}{1 - p} \] which is finite. Thus, \[ I = \frac{1}{p-1} \] which is finite for \(p > 1\). - For \(p \leq 1\): \[ I \text{ diverges} \] because: - At \(p=1\): \[ I = \int_1^\infty \frac{1}{x} dx = \infty \] - At \(p < 1\): \[ I = \int_1^\infty x^{-p} dx \text{ diverges even faster} \] **Conclusion:** \[ \boxed{ \text{The series } \sum_{n=1}^\infty \frac{1}{n^p} \text{ converges iff } p > 1 } \] --- #### **(b) Evaluate the integral \(\int_1^\infty \frac{1}{x^p} dx\) or show that it diverges.** We have already computed this in part (a). - **If \(p > 1\):** \[ \int_1^\infty \frac{1}{x^p} dx = \frac{1}{p - 1} \] which **converges**. - **If \(p \leq 1\):** The integral **diverges**. --- #### **(c) Consider the power series \(\sum_{n=0}^\infty \frac{z^n}{n!}\). Show that the series converges for every \(z \in \mathbb{R}\).** This is the exponential series: \[ e^z = \sum_{n=0}^\infty \frac{z^n}{n!} \] **Step 1: Use the ratio test** \[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{z^{n+1}/(n+1)!}{z^n / n!} \right| = \lim_{n \to \infty} \left| \frac{z}{n+1} \right| = 0 \] since \(n \to \infty\), the ratio tends to zero for all real \(z\). **Step 2: Conclusion** Because the ratio test yields a limit of zero (less than 1), the series **converges for all real \(z\)**. --- #### **(d) Consider the integral \(\int_1^\infty \frac{1}{x^p} dx\). Show that it converges, by a suitable comparison or otherwise.** This is similar to part (b). We showed: - For \(p > 1\), the integral converges. - For \(p \leq 1\), it diverges. **Comparison:** - For \(x \geq 1\), \(\frac{1}{x^p} \leq \frac{1}{x^{p}}\). - When \(p > 1\), \(\frac{1}{x^p}\) is integrable over \([1, \infty)\), as previously shown. **Conclusion:** \[ \boxed{ \text{The integral converges iff } p > 1 } \] --- ### **Summary of Key Results:** - The p-series \(\sum_{n=1}^\infty \frac{1}{n^p}\) converges iff \(p > 1\). - The integral \(\int_1^\infty \frac{1}{x^p} dx\) converges iff \(p > 1\). - The exponential series converges for every real \(z\). --- Let me know if you'd like explanations of any step or to proceed further!

Certainly! Here are the final answers summarized: **Question 6:** - Left endpoint approximation: **2.75** - Right endpoint approximation: **2.75** - Trapezoidal approximation: **2.75** **Question 7:** - (a) \(21 = 21 e^{i \cdot 0}\), \(545 = 545 e^{i \cdot 0}\) - (b) Product in polar form: \(1425 e^{i \cdot 82^\circ}\) ≈ \(198.4 + 1412.8 i\) - (c) \(z \approx 198.4 + 1412.8 i\) **Question 8:** - (a) The series \(\sum_{n=1}^\infty \frac{1}{n^p}\) converges if and only if \(p > 1\). - (b) The integral \(\int_1^\infty \frac{1}{x^p} dx\) converges iff \(p > 1\). - (c) The exponential series \(\sum_{n=0}^\infty \frac{z^n}{n!}\) converges for all \(z \in \mathbb{R}\). - (d) The integral \(\int_1^\infty \frac{1}{x^p} dx\) converges iff \(p > 1\). Let me know if you need further assistance!