Let f(t) be the depth, in centimeters, of water in a tankat time t, inminutes.(a) What

In a time of t seconds, a particle moves a distance of smeters from its starting point, where s = 3t2.(a) Find the average velocity between t = 1 and t =1 + h if:(i) h = 0.1, (ii) h = 0.01, (iii) h = 0.001.(b) Use your answers to part (a) to estimate the instantaneousvelocity of the particle at time t = 1.

Figure 2.59 shows the graph of f. Match the derivativesin the table with the points a, b, c, d, e.a b c d efxFigure 2.59x f(x)00.520.52

(a) Use a graph of f(x) = 2 x3 to decide whetherf(1) is positive or negative. Give reasons.(b) Use a small interval to estimate f(1).

Estimate f(2) for f(x) = 3x. Explain your reasoning.

(a) The function f is given in Figure 2.60. At which ofthe labeled points is f(x) positive? Negative? Zero?(b) At which labeled point is f largest? At which labeledpoint is f most negative?B CDEF GAf(x)xFigure 2.60

Use Figure 2.61 to fill in the blanks in the following statementsabout the function g at point B.(a) g( ) = (b) g( ) =(1.95, 5.02)(2, 5)Bg(x)Tangent lineFigure 2.61

In a time of t seconds, a particle moves a distance of smeters from its starting point, where s = sin(2t).(a) Find the average velocity between t = 1 and t =1 + h if:(i) h = 0.1, (ii) h = 0.01, (iii) h = 0.001.(b) Use your answers to part (a) to estimate the instantaneousvelocity of the particle at time t = 1.

The distance (in feet) of an object from a point is givenby s(t) = t2, where time t is in seconds.(a) What is the average velocity of the object betweent = 3 and t = 5?(b) By using smaller and smaller intervals around 3, estimatethe instantaneous velocity at time t = 3.

Let f(x) = 5x. Use small intervals to estimate f(2).

Using slopes to left and right of 0, estimate P(0) ifP(t) = 200(1.05)t .

Sketch the graphs of the derivatives of the functions shown in Problems 1120. Be sure your sketches are consistent with the important features of the graphs of the original functions.

Sketch the graphs of the derivatives of the functions shown in Problems 1120. Be sure your sketches are consistent with the important features of the graphs of the original functions.

Sketch the graphs of the derivatives of the functions shown in Problems 1120. Be sure your sketches are consistent with the important features of the graphs of the original functions.

Sketch the graphs of the derivatives of the functions shown in Problems 1120. Be sure your sketches are consistent with the important features of the graphs of the original functions.

Sketch the graphs of the derivatives of the functions shown in Problems 1120. Be sure your sketches are consistent with the important features of the graphs of the original functions.

Sketch the graphs of the derivatives of the functions shown in Problems 1120. Be sure your sketches are consistent with the important features of the graphs of the original functions.

Sketch the graphs of the derivatives of the functions shown in Problems 1120. Be sure your sketches are consistent with the important features of the graphs of the original functions.

Sketch the graphs of the derivatives of the functions shown in Problems 1120. Be sure your sketches are consistent with the important features of the graphs of the original functions.

Sketch the graphs of the derivatives of the functions shown in Problems 1120. Be sure your sketches are consistent with the important features of the graphs of the original functions.

Sketch the graphs of the derivatives of the functions shown in Problems 1120. Be sure your sketches are consistent with the important features of the graphs of the original functions.

(a) Estimate f(2) using the values of f in the table.(b) For what values of x does f(x) appear to be positive?Negative?x 0 2 4 6 8 10 12f(x) 10 18 24 21 20 18 15

Figure 2.62 is the graph of f, the derivative of a functionf. On what interval(s) is the function f(a) Increasing? (b) Decreasing?x1x2x3 x4 x5fxFigure 2.62: Graph of f, not f

The average weight W of an oak tree in kilograms that isx meters tall is given by the function W = f(x). Whatare the units of measurement of f(x)?

The percent, P, of US households with a personal computeris a function of the number of years, t, since 1982(when the percent was essentially zero), so P = f(t).Interpret the statements f(20) = 57 and f(20) = 3.

The temperature, T , in degrees Fahrenheit, of a cold yamplaced in a hot oven is given by T = f(t), where t is thetime in minutes since the yam was put in the oven.(a) What is the sign of f(t)? Why?(b) What are the units of f(20)? What is the practicalmeaning of the statement f(20) = 2?

The quantity sold, q, of a certain product is a function ofthe price, p, so q = f(p). Interpret each of the followingstatements in terms of demand for the product:(a) f(15) = 200 (b) f(15) = 25.

Figure 2.63 shows the length, L, in cm, of a sturgeon (atype of fish) as a function of the time, t, in years.29 Estimatef(10). Give units and interpret your answer.5 10 15 2050100150f(t)t (years)length (cm)Figure 2.63

Suppose that f(x) is a function with f(20) = 345 andf(20) = 6. Estimate f(22).

The wind speed W in meters per second at a distance xkilometers from the center of a hurricane is given by thefunction W = h(x). What does the fact that h(15) > 0tell you about the hurricane?

You drop a rock from a high tower. After it falls x metersits speed S in meters per second is S = h(x). What isthe meaning of h(20) = 0.5 ?

If t is the number of years since 2011, the population, P,of China, in billions, can be approximated by the functionP = f(t) = 1.34(1.004)t .Estimate f(9) and f(9), giving units.What do these twonumbers tell you about the population of China?

After investing $1000 at an annual interest rate of 7%compounded continuously for t years, your balance is$B, where B = f(t). What are the units of dB/dt?What is the financial interpretation of dB/dt?

Let f(t) be the depth, in centimeters, of water in a tankat time t, inminutes.(a) What does the sign of f(t) tell us?(b) Explain the meaning of f(30) = 20. Include units.(c) Using the information in part (b), at time t = 30minutes, find the rate of change of depth, in meters,with respect to time in hours. Give units.

For the function graphed in Figure 2.64, are the followingnonzero quantities positive or negative?(a) f(2) (b) f(2) (c) f(2)2411xf(x)Figure 2.64

At one of the labeled points on the graph in Figure 2.65both dy/dx and d2y/dx2 are positive. Which is it?A BCD EyxFigure 2.65

Aclimber onMount Everest is 6000 meters from the startof a trail and at elevation 8000 meters above sea level. Atx meters from the start, the elevation of the trail is h(x)meters above sea level. If h(x) = 0.5 for x near 6000,what is the approximate elevation another 3 meters alongthe trail?

Let f(v) be the gas consumption (in liters/km) of a cargoing at velocity v (in km/hr). In other words, f(v) tellsyou how many liters of gas the car uses to go one kilometerat velocity v. Explain what the following statementstell you about gas consumption:f(80) = 0.05 and f(80) = 0.0005.

To study traffic flow, a city installs a device which recordsC(t), the total number of cars that have passed by t hoursafter 4:00 am. The graph of C(t) is in Figure 2.66.(a) When is the traffic flow the greatest?(b) Estimate C(2).(c) What does C(2) mean in practical terms?2 4 6100030005000tC(t)Figure 2.66

The temperature, H, in degrees Celsius, of a cup of coffeeplaced on the kitchen counter is given by H = f(t),where t is in minutes since the coffee was put on thecounter.(a) Is f(t) positive or negative? Give a reason for youranswer.(b) What are the units of f(20)? What is its practicalmeaning in terms of the temperature of the coffee?

Suppose that f(x) is a function with f(100) = 35 andf(100) = 3. Estimate f(102).

Suppose P(t) is the monthly payment, in dollars, on amortgage which will take t years to pay off.What are theunits of P(t)? What is the practical meaning of P(t)?What is its sign?

The population, P, of a city (in thousands) at time t (inyears) is P = 700e0.035t. Estimate the relative rate ofchange of the population at t = 3 using(a) t = 1 (b) t = 0.1 (c) t = 0.01

For each part below, sketch a graph of a function thatsatisfies the given conditions. There are many possiblecorrect answers.(a) f > 0 and f > 0.(b) f > 0 and f < 0.(c) f < 0 and f > 0.(d) f < 0 and f < 0.

For each function, do the signs of the first and secondderivatives of the function appear to be positive or negativeover the given interval?(a)x 1.0 1.1 1.2 1.3 1.4 1.5f(x) 10.1 11.2 13.7 16.0 21.2 27.7(b)x 1.0 1.1 1.2 1.3 1.4 1.5f(x) 10.1 9.9 8.1 6.0 3.9 0.1(c)x 1.0 1.1 1.2 1.3 1.4 1.5f(x) 1000 1010 1015 1018 1020 1021

The length L of the day in minutes (sunrise to sunset) xkilometers north of the equator on June 21 is given byL = f(x). What are the units of(a) f(3000)? (b) f(3000)?

Sketch the graph of a function whose first and secondderivatives are everywhere positive.

At which of the marked x-values in Figure 2.67 can thefollowing statements be true?(a) f(x) < 0(b) f(x) < 0(c) f(x) is decreasing(d) f(x) is decreasing(e) Slope of f(x) is positive(f) Slope of f(x) is increasingx1 x2 x3x4 x5f(x)xFigure 2.67

Sketch the graph of the height of a particle against timeif velocity is positive and acceleration is negative.

Figure 2.68 gives the position, f(t), of a particle at timet. At which of the marked values of t can the followingstatements be true?(a) The position is positive(b) The velocity is positive(c) The acceleration is positive(d) The position is decreasing(e) The velocity is decreasingt1 t2t3 t4 t5f(t)tFigure 2.68

A high school principal is concerned about the drop inthe percentage of students who graduate from her school,shown in the following table, compared with the fall innational average from69.2% in 2006 to 68.8% in 2007.30Year entered school, t 1995 1998 2001 2004 2007Percent graduating, P 62.4 54.1 48.0 43.5 41.8(a) Calculate the average rate of change of P for eachof the three-year intervals between 1995 and 2007.(b) Does d2P/dt2 appear to be positive or negative between1995 and 2007?(c) Explain why the values of P and dP/dt are troublesometo the principal.(d) Explain why the sign of d2P/dt2 and the magnitudeof dP/dt in the year 2004 may give the principalsome cause for optimism.

Students were asked to evaluate f(4) from the followingtable which shows values of the function f:x 1 2 3 4 5 6f(x) 4.2 4.1 4.2 4.5 5.0 5.7 Student A estimated the derivative as f(4) f(5) f(4)5 4= 0.5. Student B estimated the derivative as f(4) f(4) f(3)4 3= 0.3. Student C suggested that they should split the differenceand estimate the average of these two results,that is, f(4) 12 (0.5 + 0.3) = 0.4.(a) Sketch the graph of f, and indicate how the threeestimates are represented on the graph.(b) Explain which answer is likely to be best.

Figure 2.69 shows the rate at which energy, f(v), is consumedby a bird flying at speed v meters/sec.(a) What rate of energy consumption is needed by thebird to keep aloft, without moving forward?(b) What does the shape of the graph tell you about howbirds fly?(c) Sketch f(v).

If a quantity is 100 and has relative rate of change 3%per minute, we expect the quantity to be about 97 oneminute later.

If a population of animals changes from 500 to 400 inone month, then the relative rate of change is about 100animals per month.

Linear functions have constant rate of change and exponentialfunctions have constant relative rate of change.