Find the derivative of \(f(x) = \ln(x^2 + 1)\) at \(x = 1\).
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- e 2\). Simplify and find \(\lim_{x o 2} f(x)\).
- Factor numerator: \(x^2 - 4 = (x - 2)(x + 2)\), so \(f(x) = rac{(x - 2)(x + 2)}{x - 2} = x + 2\) for \(x
- Thus, \(\lim_{x o 2} f(x) = 2 + 2 = 4\).
- Use chain rule: \(f'(x) = rac{1}{x^2 + 1} \cdot 2x = rac{2x}{x^2 + 1}\).
- At \(x = 1\): \(f'(1) = rac{2(1)}{1 + 1} = rac{2}{2} = 1\).
- The sum of the first \(n\) terms of an arithmetic sequence is \(S_n = 3n^2 + 5n\). Find the 10th term.
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A rectangular garden has a length of 15 meters and a width that is 3 meters less than the length. If the garden is expanded by increasing both dimensions by 2 meters, what is the new area of the garden?
Original width = 15 - 3 = 12 meters
New length = 15 + 2 = 17 meters
New width = 12 + 2 = 14 meters
New area = 17 meters × 14 meters = 238 square meters
A train travels at a speed of 60 km/h for 2 hours, then increases its speed to 90 km/h for the next 1.5 hours. What is the total distance traveled?
Distance at 60 km/h = 60 km/h × 2 hours = 120 km