Summer Practice Questions

Question 0

The surface area of a cube (NOT the volume) changes when the length of an edge changes. The relationship that links surface area to edge length is A=6e^2.

What general relationship links the rate of change in the edge and the rate of change in the surface area?

Answer 0

A=6e^2
d/dt (A)= d/dt (6e^2)
dA/dt= 2•6e•de/dt
dA/dt= 12e•de/dt

Question 1

Solve using substitution.

∫sin(4兀t) dt

Answer 1

Let u= 4兀t, so du=4兀 dt. Rearrange problem so (1/4兀) = dt.
(1/4兀) ∫sin(u) du= -(1/4兀)cos(4兀t)+C

Final answer: -(1/4兀)•cos(4兀t) + C

Question 2

Solve implicitly.

x^2 + y^2 = xy

Answer 2

Take the derivative of both sides (the whole thing). Use the chain rule for the second part of the equation on the left.

Final answer: 2x + 2yy’ = xy’ + y

Question 3

A car is travelling 50 mph (77ft/s) when the brakes are fully applied, producing a constant decelleration of 22ft/s^2. What is the distance travelled before the car stops?

Answer 3

a= -22= v'(t)
v(t)= -22t + C; v1= 77ft/s
s(t)= -11t^2+ 77t+ C; s1= 0

The car stops when v(t)= 0. So:
0= -22(t) +77; solving for t, we find t= 77/22, or 7/2.

Now, solving for distance, we have:
s(t)= -11(7/2)^2 + 77(7/2). Simplify.

Final answer: 134.75 ft