(A) Acceleration is the derivative of velocity. Since the velocity is linear,
its derivative is its slope.
(C) Positive velocity implies motion to the right (t < 2); negative velocity
(t > 2) implies motion to the left.
(B) The average rate of change of velocity is .
(E) The slope of y = x^3 is 3x^2 . It is equal to 3 when x = ±1. At x = 1, the
equation of the tangent is
y − 1 = 3(x − 1) or y = 3x + 2.
At x = −1, the equation is
y + 1 = 3(x + 1) or y = 3x + 2.
(C) Let the tangent to the parabola from (3, 5) meet the curve at (x 1 , y 1 ).
Its equation is y − 5 = 2x 1 (x − 3). Since the point (x 1 , y 1 ) is on both the
tangent and the parabola, we solve simultaneously:
y 1 − 5 = 2x 1 (x 1 − 3) and y 1 = x 12
The points of tangency are (5, 25) and (1, 1). The slopes, which equal
2 x 1 , are 10 and 2.
(E).
