Given r(2) = 4, s(2) = 1, s(4) = 2, r(2) = 1, s(2) = 3, and s(4) = 3, compute the

Find the derivatives for the functions in Problems 140. Assume k is a constant. f(t) = 6t4

Find the derivatives for the functions in Problems 140. Assume k is a constant. f(x) = x3 3x2 + 5x

Find the derivatives for the functions in Problems 140. Assume k is a constant. P(t) = e2t

Find the derivatives for the functions in Problems 140. Assume k is a constant. W = r3 + 5r 12

Find the derivatives for the functions in Problems 140. Assume k is a constant. C = e0.08q

Find the derivatives for the functions in Problems 140. Assume k is a constant. y = 5e0.2t

Find the derivatives for the functions in Problems 140. Assume k is a constant. y = xe3x

Find the derivatives for the functions in Problems 140. Assume k is a constant. s(t) = (t2 +4)(5t 1)

Find the derivatives for the functions in Problems 140. Assume k is a constant. g(t) = e(1+3t)2

Find the derivatives for the functions in Problems 140. Assume k is a constant. f(x) = x2 +3lnx

Find the derivatives for the functions in Problems 140. Assume k is a constant. Q(t) = 5t + 3e1.2t

Find the derivatives for the functions in Problems 140. Assume k is a constant. g(z) = (z2 + 5)3

Find the derivatives for the functions in Problems 140. Assume k is a constant. f(x) = 6(5x 1)3

Find the derivatives for the functions in Problems 140. Assume k is a constant. f(z) = ln(z2 + 1)

Find the derivatives for the functions in Problems 140. Assume k is a constant. h(x) = (1+ex)10

Find the derivatives for the functions in Problems 140. Assume k is a constant. q = 100e0.05p

Find the derivatives for the functions in Problems 140. Assume k is a constant. y = x2 ln x

Find the derivatives for the functions in Problems 140. Assume k is a constant. s(t) = t2 + 2lnt

Find the derivatives for the functions in Problems 140. Assume k is a constant. P = 4t2 + 7sint

Find the derivatives for the functions in Problems 140. Assume k is a constant. R(t) = (sint)5

Find the derivatives for the functions in Problems 140. Assume k is a constant. h(t) = ln&et t'

Find the derivatives for the functions in Problems 140. Assume k is a constant. f(x) = sin(2x)

Find the derivatives for the functions in Problems 140. Assume k is a constant. g(x) = 25x2 ex

Find the derivatives for the functions in Problems 140. Assume k is a constant. h(t) = t + 4 t 4

Find the derivatives for the functions in Problems 140. Assume k is a constant. y = x2 cos x

Find the derivatives for the functions in Problems 140. Assume k is a constant. h(x) = ln(1+ex)

Find the derivatives for the functions in Problems 140. Assume k is a constant. h(w) = elnw+1

Find the derivatives for the functions in Problems 140. Assume k is a constant. h(x) = ln(x3 + x)

Find the derivatives for the functions in Problems 140. Assume k is a constant. q(x) = 1 +ex 1 ex

Find the derivatives for the functions in Problems 140. Assume k is a constant. q(x) = x 1 + x

Find the derivatives for the functions in Problems 140. Assume k is a constant. f(x) = xex

Find the derivatives for the functions in Problems 140. Assume k is a constant. h(x) = sin(ex)

Find the derivatives for the functions in Problems 140. Assume k is a constant. h(x) = cos(x3)

Find the derivatives for the functions in Problems 140. Assume k is a constant. z = 3t + 1 5t + 2

Find the derivatives for the functions in Problems 140. Assume k is a constant. z = t2 + 5t + 2 t + 3

Find the derivatives for the functions in Problems 140. Assume k is a constant. h(p) = 1 + p2 3 +2p2

Find the derivatives for the functions in Problems 140. Assume k is a constant. y = 3x 3 + 33 x

Find the derivatives for the functions in Problems 140. Assume k is a constant. f(t) = sinet + 1

Find the derivatives for the functions in Problems 140. Assume k is a constant. f(t) = 7 ekt

Find the derivatives for the functions in Problems 140. Assume k is a constant. g(t) = ln(kt) + t ln(kt) t

Let f(x) = x2+1. Compute the derivatives f(0), f(1), f(2), and f(1). Check your answers graphically.

Let f(x) = x2 +3x 5. Find f(0), f(3), f(2).

Find the equation of the line tangent to the graph of f at (1, 1), where f is given by f(x) = 2x3 2x2 + 1.

Some antique furniture increased very rapidly in price over the past decade. For example, the price of a particular rocking chair is well approximated by V = 75(1.35)t , where V is in dollars and t is in years since 2000. Find the rate, in dollars per year, at which the price is increasing at time t.

According to the US Census, the world population P, in billions, was approximately10 P = 6.8e0.012t where t is in years since January 1, 2009. At what rate was the worlds population increasing on that date? Give your answer in millions of people per year.

A football player kicks a ball at an angle of 30 from the ground with an initial velocity of 64 feet per second. Its height t seconds later is h(t) = 32t 16t2. (a) Graph the height of the ball as a function of the time. (b) Find v(t), the velocity of the ball at time t. (c) Find v(1), and explain what is happening to the ball at this time. What is the height of the ball at this time?

The graph of y = x3 9x2 16x + 1 has a slope of 5 at two points. Find the coordinates of the points.

(a) Find the slope of the graph of f(x) = 1 ex at the point where it crosses the x-axis. (b) Find the equation of the tangent line to the curve at this point.

Find the equation of the tangent line to the graph of P(t) = t ln t at t = 2. Graph the function P(t) and the tangent line Q(t) on the same axes.

The balance, $B, in a bank account t years after a deposit of $5000 is given by B = 5000e0.08t. At what rate is the balance in the account changing at t = 5 years? Use units to interpret your answer in financial terms.

If a cup of coffee is left on a counter top, it cools off slowly. The temperature in degrees Fahrenheit t minutes after it was left on the counter is given by C(t) = 74+103e0.033t . (a) What was the temperature of the coffee when it was left on the counter? (b) If the coffee was left on the counter for a long time, what is the lowest temperature the coffee would reach? What does that temperature represent? (c) Find C(5) and C(5). Explain what this tells about the temperature of the coffee. (d) Without calculation, decide if the magnitude of C(50) is greater or less than the magnitude of C(5)? Why?

With length, l, in meters, the period T , in seconds, of a pendulum is given by T = 23 l 9.8 . (a) How fast does the period increase as l increases? (b) Does this rate of change increase or decrease as l increases?

One gram of radioactive carbon-14 decays according to the formula Q = e0.000121t , where Q is the number of grams of carbon-14 remaining after t years. (a) Find the rate at which carbon-14 is decaying (in grams/year). (b) Sketch the rate you found in part (a) against time.

The temperature, H, in degrees Fahrenheit (F), of a can of soda that is put into a refrigerator to cool is given as a function of time, t, in hours, by H = 40+30e2t. (a) Find the rate at which the temperature of the soda is changing (in F/hour). (b) What is the sign of dH/dt? Explain. (c) When, for t 0, is themagnitude of dH/dt largest? In terms of the can of soda, why is this?

Explain for which values of a the function ax is increasing and for which values it is decreasing. Use the fact that, for a > 0, d dx (ax) = (lna)ax.

Worldwide production of solar power, in megawatts, can be modeled by f(t) = 1040(1.3)t, where t is years11 since 2000. Find f(0), f(0), f(15), and f(15). Give units and interpret your answers in terms of solar power.

The temperature Y in degrees Fahrenheit of a yam in a hot oven t minutes after it is placed there is given by Y (t) = 350(1 0.7e0.008t). (a) What was the temperature of the yam when it was placed in the oven? (b) What is the temperature of the oven? (c) When does the yam reach 175 F? (d) Estimate the rate at which the temperature of the yam is increasing when t = 20.

Keplers third law of planetary motion states that P2 = kd3, where P represents the time, in earth days, it takes a planet to orbit the sun once, d is the planets average distance, in miles, from the sun, and k is a constant.12 (a) If k = 1.65864 1019, write P as a function of d. (b) Mercury is approximately 36 million miles from the sun. How long does it take Mercury to orbit the sun? (c) Find P(d). What does the sign of the derivative tell us about the time it takes a planet to orbit the sun? Why does this make sense from a practical viewpoint?

Imagine you are zooming in on the graph of each of the following functions near the origin: y =x y= x y= x2 y = x3 + 1 2x2 y = x3 y = ln(x+ 1) y = 1 2 ln(x2 +1) y = 2x x2 Which of them look the same? Group together those functions which become indistinguishable near the origin, and give the equations of the lines they look like.

Given a power function of the form f(x) = axn, with f(2) = 3 and f(4) = 24, find n and a.

Given r(2) = 4, s(2) = 1, s(4) = 2, r(2) = 1, s(2) = 3, and s(4) = 3, compute the following derivatives, or state what additional information you would need to be able to compute the derivative. (a) H(2) if H(x) = r(x) + s(x) (b) H(2) if H(x) = 5s(x) (c) H(2) if H(x) = r(x) s(x) (d) H(2) if H(x) = 4r(x)

Given F(2) = 1, F(2) = 5, F(4) = 3, F(4) = 7 and G(4) = 2, G(4) = 6,G(3) = 4,G(3) = 8, find: (a) H(4) if H(x) = F(G(x)) (b) H(4) if H(x) = F(G(x)) (c) H(4) if H(x) = G(F(x)) (d) H(4) if H(x) = G(F(x))

A dose, D, of a drug causes a temperature change, T , in a patient. For C a positive constant, T is given by T = $C 2 D 3 %D2. (a) What is the rate of change of temperature change with respect to dose? (b) Forwhat doses does the temperature change increase as the dose increases?

A yamis put in a hot oven, maintained at a constant temperature 200C. At time t = 30 minutes, the temperature T of the yam is 120 and is increasing at an (instantaneous) rate of 2/min. Newtons law of cooling (or, in our case, warming) implies that the temperature at time t is given by T (t) = 200 aebt. Find a and b.

For Problems 6570, let h(x) = f(x) g(x), and k(x) = f(x)/g(x), and l(x) = g(x)/f(x). Use Figure 3.24 to estimate the derivatives. h(1)

For Problems 6570, let h(x) = f(x) g(x), and k(x) = f(x)/g(x), and l(x) = g(x)/f(x). Use Figure 3.24 to estimate the derivatives. k(1)

For Problems 6570, let h(x) = f(x) g(x), and k(x) = f(x)/g(x), and l(x) = g(x)/f(x). Use Figure 3.24 to estimate the derivatives. h(2)

For Problems 6570, let h(x) = f(x) g(x), and k(x) = f(x)/g(x), and l(x) = g(x)/f(x). Use Figure 3.24 to estimate the derivatives. k(2)

For Problems 6570, let h(x) = f(x) g(x), and k(x) = f(x)/g(x), and l(x) = g(x)/f(x). Use Figure 3.24 to estimate the derivatives. l(1)

For Problems 6570, let h(x) = f(x) g(x), and k(x) = f(x)/g(x), and l(x) = g(x)/f(x). Use Figure 3.24 to estimate the derivatives. l(2)

On what intervals is the graph of f(x) = x4 4x3 both decreasing and concave up?

Given p(x) = xn x, find the intervals over which p is a decreasing function when: (a) n = 2 (b) n = 1 2 (c) n = 1

Using the equation of the tangent line to the graph of ex at x = 0, show that ex 1 + x for all values of x. A sketch may be helpful.

In Section 1.10 the depth, y, in feet, of water in Portland, Maine is given in terms of t, the number of hours since midnight, by y = 4.9 + 4.4 cos $ 6 t%. (a) Find dy/dt. What does dy/dt represent, in terms of water level? (b) For 0 t 24, when is dy/dt zero? (Figure 1.103 may be helpful.) Explain what it means (in terms of water level) for dy/dt to be zero.

Using a graph to help you, find the equations of all lines through the origin tangent to the parabola y = x2 2x + 4. Sketch the lines on the graph.

A museum has decided to sell one of its paintings and to invest the proceeds. If the picture is sold between the years 2000 and 2020 and the money from the sale is invested in a bank account earning 5% interest per year compounded annually, then B(t), the balance in the year 2020, depends on the year, t, in which the painting is sold and the sale price P(t). If t is measured from the year 2000 so that 0 < t < 20 then B(t) = P(t)(1.05)20t. (a) Explain why B(t) is given by this formula. (b) Show that the formula for B(t) is equivalent to B(t) = (1.05)20 P(t) (1.05)t . (c) Find B(10), given that P(10) = 150,000 and P(10) = 5000.

Figure 3.25 shows the number of gallons, G, of gasoline used on a trip of M miles. (a) The function f is linear on each of the intervals 0<M < 70 and 70<M < 100. What is the slope of these lines? What are the units of these slopes? (b) What is gas consumption (in miles per gallon) during the first 70 miles of this trip? During the next 30 miles? (c) Figure 3.26 shows distance traveled, M (in miles), as a function of time t, in hours since the start of the trip. Describe this trip in words. Give a possible explanation for what happens one hour into the trip. What do your answers to part (b) tell you about the trip? (d) If we let G = k(t) = f(h(t)), estimate k(0.5) and interpret your answer in terms of the trip. (e) Find k(0.5) and k(1.5). Give units and interpret your answers.

The 2010 Census13 determined the population of the US was 308.75 million on April 1, 2010. If the population was increasing exponentially at a rate of 2.85 million per year on that date, find a formula for the population as a function of time, t, in years since that date.

The speed of sound in dry air is f(T) = 331.331 + T 273.15 meters/second where T is the temperature in degrees Celsius. Find a linear function that approximates the speed of sound for temperatures near 0C.

Global temperatures have been rising, on average, for more than a century, sparking concern that the polar ice will melt and sea levels will rise. With t in years since 1880, fitting functions to the data14 gives three models for the average global temperature in Celsius: f(t) = 13.625 + 0.006t g(t) = 0.00006t2 0.0017t + 13.788 h(t) = 13.63e0.0004t . (a) What family of functions is used in each model? (b) Find the rate of change of temperature in 2010 in each of the three models. Give units. (c) For each model, find the change in temperature over a 130-year period if the temperature had been changing at the rate you found in part (b) for that model. (d) For each model, find the predicted change in temperature for the 130 years from 1880 to 2010. (e) For which model, if any, are the answers equal in parts (c) and (d)? (f) For which model is the discrepancy largest between the answers in parts (c) and (d)?