Solution for Explore! on Page 374

At x = 1, the slopes for each curve appear equal. Every tangent line at x = 1 has a slope of 3, although each would have differing y-intercepts.

When x = 1, the F’(x) has a value of approximately 1, which is equal to f(1). (The numerical derivative is used to obtain F’(x) explaining negligible differences in values.) When x = -2, F’(x) = -.5, again equal to f(-2). Choose any other x-values and verify that F’(x) = f(x).

Place the integers -5 to 5 in L1 (STAT EDIT), then trace to x = 2. Use the up arrow key to search for the antiderivative which passes through (2,6). This appears as F(x) = x³ + x - 4, supporting the analytical solution. For f(x) = 3x² - 2, the same window produces the graph on the right, with F(x) = x³ - 2x + 2, which should be confirmed algebraically as in Example 1.3.

The TRACE key helps find that the population will be 5126 people in 9 months. Setting Y2 = 6000, either trace to the intersection point or use the intersection feature of the graphing calculator to determine that it takes about 37 2/3 months to reach the population plateau of 6000 people.

The car stops when zero velocity is reached at (3,99). Note that, although the parabola curves back, there is no corresponding physical interpretation to the part of the curve for x > 3, since the car has stopped for good. For the initial velocity, v(0) = 60 mph (88 fts), it takes 4 sec for the car to stop at a distance of 176 ft. When x = 3 sec, the car is still traveling at a rate of 22 feet per second.

The family of solution curves, symmetric to the y-axis, is displayed below. It appears graphically that the curve corresponding to constant value C = 8 in the list L₁ passes through the point (0, 2).

Any value of x > 5.2 will produce a p(x) < 2.01.