[High School AP Calc] a few problems

[insanefreddy926]

yo here's a few take home questions I got. All help is appreciated. :]

1. An object moves along the x-axis with initial position x(0)=2. The velocity of the object at time t >= 0 is given by v(t)=sin((pi/3)t). What is the acceleration of the object at time t=4? What is the position of the object at time t=4? What is the total distance traveled by the object over the time interval 0 <= t <= 4?

I got:

a(4) = -pi/6
x(4) = (4pi + 9)/(2pi)
total distance = 15/(2pi)


2. Let f(x)= x^3 + px^2 + qx. Find the values of p and q so that f(-1)=-8 and f'(-1)=12. Find the value of p so that the graph of f changes concavity at x=2. Under what conditions of p and q will the graph of f be increasing everywhere?

for the first part I got p = -2 and q = 5
for the second part I got p = -6
for the last part I said whenever p is greater than or equal to 0 and q is greater than 0


3. Let f(x)= 4 - x^2. For 0 <= w <= 2, let A(w) be the area of the triangle formed by the coordinate axes and the line tangent to the graph of f at the point (w, 4 - w^2). Find A(1). For what value of w is A(w) a minimum?

I know what's going on here, but I'm having trouble coming up with the equation for A(w).


4. Consider the curve given by the equation y^3 - 3xy = 2. Find dy/dx. Write an equation for the line normal to the curve at the point (1,2). What is the concavity of the curve at that point?

edit: whoops I messed up, the dy/dx value I got is y/(y^2 - x) and the normal line is y - 2 = (-3/2)(x - 1)
don't know how to determine the concavity there. would it be from the second derivative?