The top of a ladder slides down a vertical wall at a rate

Question

CHAPTER 3-

Problem 1- Find the points on the curve y = 2x³ + 3x² - 12x + 1 where the tangent line is horizontal.

Problem 2- Show that the curve y = 2e^x + 3x + 5x³ has no tangent line with slope 2.

Problem 3- Let

Is f differentiable at x = 1? Sketch the graphs of f and f'.

Problem 4- Find equations of the tangent line and normal line to the curve y = 2xe^x at the point (0, 0).

Problem 5- Find R'(0), where

R(x) = x - 3x³ + 5x⁵/1 + 3x³ + 6x⁶ + 9x⁹.

Problem 6- If g is a twice-differentiable function and f(x) = xg(x²), find f'' in terms of g, g', g''.

Problem 7- If f(x) + x²(f(x))³ = 10 and f(1) = 2, find f'(1).

Problem 8- Find y' if y = ln(x² + y²).

Problem 9- Find d⁹/dx⁹ (x⁸ ln x).

Problem 10- A sample of tritium-3 decayed to 94.5% of its original amount after a year. What is the half-life of tritium-3? How long would it take the sample to decay to 20% of its original amount?

Problem 11- Suppose y = √(2x + 1), where x and y are functions of t. If dx/dt = 3, find dy/dt when x = 4.

Problem 12- The top of a ladder slides down a vertical wall at a rate of 0.15 m/s. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s. How long is the ladder?