What is the linear approximation to f(x) = 2cos x, near x = pi/2?

Question

1. What is the linear approximation to f(x) = 2cos x, near x = pi/2?

 

2. What is the linear approximation to f(x) = 5 ln (1 + x)² at (0, 0)?

 

3. What is the derivative of f(x) = e^x ln x?

 

4. How long will it take $5,000 to compound continuously to $30,000 at 8% interest? Choose the closest answer.

 

5. Where is f(x) = 2x³ + 12x² + 3x + 7 concave down?

Answer 1

1. f(x) ≈ L(x) = f(π/2) + f'(π/2)(x-π/2)

 

                   = 2cos(π/2) - 2sin(π/2)(x - π/2)

 

                   = 2(0) - 2(1)(x-π/2)

 

                   = 2x + π

 

2. f(x) ≈ L(x) = f(0) + f'(0)(x - 0)

 

                   = 5ln1 + 10x

 

                   = 10x

 

3.  Use the product rule

 

4. Solve the equation 30000 = 5000e^0.08t for t

 

5. Determine where f"(x) < 0