Midterm 2 review worksheet find the equation of the tangent

Question

Midterm 2 Review Worksheet - Math 1A, section 103

1. What is the derivative of f(x) = (x + 1)²?

2. Find the derivative of cos(x)^sin(x).

3. Find the equation of the tangent line to the ellipse x² + 2y² = 3 at the point (1, 1). Sketch the ellipse and the tangent line.

4. If f(x) = sin(2x), what is the nth derivative dⁿ/dxⁿ f(x)?

5. Use linear approximation or differentials to approximate the value of √9.2. Then use your calculator to see how close your approximation is.

6. A Ferris wheel with a radius of 10 m is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is 16 m above ground level?

7. Use implicit differentiation to find the slope of the tangent line to the curve 2^y + xy = x² at the point (2, 1).

8. A freshly brewed cup of coffee has temperature 95^oC in a room at a fixed room temperature of 20^oC. When its temperature is 70^oC, it is cooling at a rate of 1^oC per minute. When does this occur? (You may use Newton's Law of Cooling: that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings: so T(t) - T_room = Ae^-kt for some A and k.)