Applications of Derivatives

The population of a town x years after 2000 is modeled by P(x).  What is the best interpretation of P'(10) = 50.

• In 2010, the population was increasing 50 people per year.
• In 2010, the population was 50 people.
• It took 10 years for the population to increase by 50 people.
• The average change in population over the 10 years was 50 people per year.

Tags

CCSS.HSF.LE.B.5

The function f is twice differentiable with f(2) = 1, f'(2)=4, and f''(2)=3.  What is the value of the approximation of f(1.9) using the line tangent to the graph of f at x = 2?

• 0.4
• 0.6
• 0.7
• 1.4

Tags

CCSS.HSF.IF.A.1

CCSS.HSF.IF.A.2

Suppose that for a twice differentiable function f (x), f '(x) > 0 and f ''(x) > 0 on the interval (1,3). If we use a linear approximation at x = 2 to estimate f (1.9), will that estimate be an underestimate or overestimate, and why?

• Underestimate since f '' > 0 on (1, 3)

• Overestimate since f '' > 0 on (1,3)

• Underestimate since f ' > 0 on (1, 3)

• Overestimate since f ' > 0 on (1, 3)

A railroad track and a road cross at right angles.  An observer stands on the road 70 meters south of the crossing and watches an eastbound train traveling at 60 meters per second.  At how many meters per second is the train moving away from the observer 4 seconds after it passes through the intersection?

• 57.60
• 57.88
• 59.20
• 67.40

The radius of a circle is increasing at a constant rate of 0.2 meters per second.  What is the rate of increase in the area of the circle at the instant when the circumference of the circle is 20π meters?

• 0.04π m²/sec
• 0.4π m²/sec
• 4π m²/sec
• 20π m²/sec

A spherical snowball melts so that its radius decreases at a rate of 4 in/sec. At what rate is the volume of the snowball changing when the radius is 4 in?

• -262π in³/sec

• -247π in³/sec

• -256π in³/sec

• -263π in³/sec

A 5 ft ladder is leaning against a wall and sliding towards the floor. The top of the ladder is sliding down the wall at a rate of 2 ft/sec. How fast is the base of the ladder sliding away from the wall when the base of the ladder is 3 ft from the wall?

• 4/3 ft/sec

• 8/7 ft/sec

• 1 ft/sec

• 8/3 ft/sec

A water tank, shaped like an inverted circular cone, has a base radius of 6 ft and a height of 9 ft. The tank is completely full and needs to be drained. The valve is opened and the water begins to decrease at a rate of 2 ft³/sec.  How fast is the height of the water changing when the water is 2 ft deep?

• -9/(8pi) ft/sec
• 9/(8pi) ft/sec
• -8/(9pi) ft/sec
• 8/(9pi) f/tsec

What is the limit?

• DNE
• 2/3
• 1/4
• Infinity

• 2

• -2

• 1/2

• -1/2