## Advanced Placement Calculus AB Practice

### Section II Part B

A graphing calculator is not allowed for the following problems.
1.Write an equation for the line tangent to the curve x

^{2} + 4x + y

^{3} = 4 at the point (1, -1).

2. dy/dx = (x + 1)

^{2}y

Sketch a slope field for the given differential equation at the points indicated.

3. dy/dx = (x + 1)

^{2}y

Let y = f(x) be the solution of this equation with the initial condition f(1) = 1. What is the equation of the tangent to the graph of f at x = 1?

4. dy/dx = (x + 1)

^{2}y

Find the solution y = f(x) to the given differential equation with the initial condition f(1) = 1.

5. Consider the closed curve in the xy plane given by 2x

^{2} + 5x + y

^{3} + 4y

^{2} + y = 8.

Show that dy/dx = -(4x + 5)/(3y

^{2} + 8y + 1).

6.

The graph of function f is shown above. Let:

Calculate g(3). Does g have a relative minimum or a relative maximum at x = -1? Calculate g"(0).

7. Let f be the function given by f(x) = (2x

^{2} + 5x -1)

^{7}.

Write the equation for the line tangent to f(x) in x = 0.

8. Find the coordinates of the two points on the x

^{2} - 2x + 4y

^{2} + 16y + 1 = 0 closed curve where the line tangent to the curve is vertical.

Verify your answers.