The graph of fat storage is linear for four weeks, then becomes concave up. Thus, the derivative of fat storage is constant for four weeks, then increases. This matches graph I. The graph of protein storage is concave up for three weeks, then becomes concave down. Thus, the derivative of protein storage is increasing for three weeks and then becomes decreasing. This matches graph II. At this point, the increases in advertising expenditures just pay for themselves. This is not sustainable since we are using more gallons than we are producing.
This might be sustainable since we are able to use the gasoline to produce many more gallons of biofuel. The number of active Facebook users at the end of April was million. The number was increasing at Thus the ozone hole is predicted to recover by At all the other points one or both of the derivatives could not be negative.
The derivative is positive on those intervals where the function is increasing and negative on those intervals where the function is decreasing. The second derivative is positive on those intervals where the graph of the function is concave up and negative on those intervals where the graph of the function is concave down.
Therefore, the second derivative is positive on the interval 0. The derivative of w t appears to be negative since the function is decreasing over the interval given. The second derivative, however, appears to be positive since the function is concave up, i. The graph must be everywhere decreasing and concave up on some intervals and concave down on other intervals. One possibility is shown in Figure 2.
Since all advertising campaigns are assumed to produce an increase in sales, a graph of sales against time would be expected to have a positive slope. A positive second derivative means the rate at which sales are increasing is increasing. If a positive second derivative is observed during a new campaign, it is reasonable to conclude that this increase in the rate sales are increasing is caused by the new campaign—which is therefore judged a success.
A negative second derivative means a decrease in the rate at which sales are increasing, and therefore suggests the new campaign is a failure. For example, between and , the rate of change is The number of passenger cars in the US was increasing at a rate of about , cars per year in This graph is increasing for all x, and is concave down to the left of 2 and concave up to the right of 2.
One possible answer is shown in Figure 2. Many other answers are also possible. The positive first derivative tells us that the temperature is increasing; the negative second derivative tells us that the rate of increase of the temperature is slowing. Thus, the temperature rose all day. Therefore, f 7 must be smaller than 24, meaning 22 is the only possible value for f 7 from among the choices given. The industry will say that the rate of discharge is increasing less quickly, and may soon level off or even start to fall.
The industry will say that the rate of discharge has decreased significantly. So the derivative of utility is positive, but the second derivative of utility is negative. So sea level rise is between mm and mm. Drawing in the tangent line at the point , R , we get Figure 2. Therefore, marginal cost at q is the slope of the graph of C q at q.
We have. The marginal cost is smallest at the point where the derivative of the function is smallest. The slope of the revenue curve is greater than the slope of the cost curve at both q1 and q2 , so the marginal revenue is greater at both production levels. Since the company will lose money, it should not produce the st item. Thus, decreasing production 0. Since increasing production increases profit, the company should increase production.
Since increasing production reduces the profit, the company should decrease production. For each q, we calculate the average rate of change of the cost and the revenue over the interval to the right. For each value of q, we calculate the average rate of change of the cost and the revenue over the interval to the right. Solutions for Chapter 2 Review 1. See Table 2. Using the intervals 0. Any small interval around 2 gives a reasonable answer. This occurs at point G.
This occurs at point A. In this case, more estimates with smaller values of h would be very helpful in making a better estimate. Also as x gets large, the graph of f x gets more and more horizontal. Alternately, we could use the interval to the left of 2, or we could use both and average the results.
Note that we are considering the average temperature of the yam, since its temperature is different at different points inside it. Moving away slightly from the center of the hurricane from a point 15 kilometers from the center moves you to a point with stronger winds. For example, the wind is stronger at After falling 20 meters the speed of the rock is increasing at a rate of 0.
We can interpret dB as the extra money added to your balance in dt years. At all the other points one or both of the derivatives could not be positive. The elevation increases approximately 1. The new elevation is about This means that a 1-kilometer increase in speed results in an increase in consumption of about 0.
At higher speeds, the vehicle burns more gasoline per km traveled than at lower speeds. It is greatest where the graph of the function C t is the steepest and increasing. The function is everywhere increasing and concave up. At x4 and x5 , because the graph is below the x-axis there. At x3 and x4 , because the graph is sloping down there. This is the same condition as part b.
At x2 and x3 , because the graph is bending downward there. At x1 , x2 , and x5 , because the graph is sloping upward there. At x1 , x4 , and x5 , because the graph is bending upward there. At t3 , t4 , and t5 , because the graph is above the t-axis there. At t2 and t3 , because the graph is sloping up there. At t1 , t2 , and t5 , because the graph is concave up there At t1 , t4 , and t5 , because the graph is sloping down there.
At t3 and t4 , because the graph is concave down there. This tells us that the bird expends more energy per second to remain still than to travel at slow speeds say 0. The upward concavity of the graph tells us that as the bird speeds up, it uses energy at a faster and faster rate.
Other answers are possible. True, this is the definition of the derivative. False, the average rate of change between 0 and 1 is given by the slope of the line connecting the points 0, f 0 and 1, f 1. False, the function r appears to be increasing, and therefore would have a positive derivative.
False, R w is increasing for all w so the derivative can not be negative at any point. True, consider any constant function. False, the opposite is true: If the derivative of f is negative on an interval, then f is decreasing on that interval. True, the derivative is the slope of the curve, which is always 0 for a horizontal line. Therefore, the derivative of the function is less and less negative, so the derivative is increasing.
False, the function ln t is an increasing function, and therefore has positive derivative. True, as long as it is understood that x is the independent variable, then the two quantities are equal. True, this interpretation is as specified in the text.
True, since the derivative is the approximate change in f when the independent variable is increased by one. True, since for the first 10 years, height is an increasing function of age. True, since if the derivative is negative, then W is a decreasing function of R. If the car is slowing down, then the derivative is decreasing, which means the second derivative is negative.
True, since ex is concave up everywhere. True, if marginal revenue is greater than marginal cost, then the amount of revenue earned on producing an additional unit will be more than the amount it costs to produce, and so it will increase the profit.
True, since the total cost function never decreases, so its derivative is never negative. False, if the revenue function is linear, then the marginal revenue is constant slope of the line, or 5. True, both the marginal revenue and the marginal cost have dollars as the dependent variable and units as the dependent.
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