Let's solve the given problem step-by-step: The function is: \[ N(x) = 1000 - \frac{3780}{x} \] ### **(a) Evaluate \( N(0) \) and \( N(37.8) \)** #### **Step 1: Calculate \( N(0) \)** \[ N(0) = 1000 - \frac{3780}{0} \] However, division by zero is undefined. So, **\( N(0) \) is undefined**. --- #### **Step 2: Calculate \( N(37.8) \)** \[ N(37.8) = 1000 - \frac{3780}{37.8} \] Calculate \( \frac{3780}{37.8} \): \[ \frac{3780}{37.8} = 100 \] So, \[ N(37.8) = 1000 - 100 = 900 \] --- ### **(b) Evaluate \( N'(x) \) and \( N'(x_f) \)** #### **Step 1: Find \( N'(x) \)** \[ N(x) = 1000 - 3780x^{-1} \] Take the derivative: \[ N'(x) = 0 - 3780 \times (-1) x^{-2} \] \[ N'(x) = 3780 x^{-2} \] \[ N'(x) = \frac{3780}{x^2} \] #### **Step 2: \( N'(x_f) \)** - \( x_f \) is not given in the question. If you have a specific value for \( x_f \), substitute it into \( N'(x) \). --- ### **(c) Evaluate \( N'(10) \) and \( N'(20) \)** #### **Step 1: Calculate \( N'(10) \)** \[ N'(10) = \frac{3780}{10^2} = \frac{3780}{100} = 37.8 \] #### **Step 2: Calculate \( N'(20) \)** \[ N'(20) = \frac{3780}{20^2} = \frac{3780}{400} = 9.45 \] --- ## **Final Answers (Summary):** **a.** - \( N(0) \) is **undefined** - \( N(37.8) = 900 \) **b.** - \( N'(x) = \frac{3780}{x^2} \) - \( N'(x_f) = \frac{3780}{x_f^2} \) (substitute value for \( x_f \) if given) **c.** - \( N'(10) = 37.8 \) - \( N'(20) = 9.45 \) Let me know if you need further explanation on any step!