Question: Suppose that x and y are related

> A baseball diamond is a 90-foot by 90-foot square. (See Fig. 9.) A player runs from first to second base at the speed of 22 feet per second. How fast is the player’s distance from third base changing when he is halfway between first and

> An airplane flying 390 feet per second at an altitude of 5000 feet flew directly over an observer. Figure 8 shows the relationship of the airplane to the observer at a later time. (a) Find an equation relating x and y. (b) Find the value of x when y is 1

> Figure 7 shows a 10-foot ladder leaning against a wall. (a) Use the Pythagorean theorem to find an equation relating x and y. (b) If the foot of the ladder is being pulled along the ground at the rate of 3 feet per second, how fast is the top end of the

> Suppose that in Boston the wholesale price, p, of oranges (in dollars per crate) and the daily supply, x (in thousands of crates), are related by the equation px + 7x + 8p = 328. If there are 4 thousand crates available today at a price of $25 per crate,

> Determine the growth constant k, then find all solutions of the given differential equation. y' – y/2 = 0

> The monthly advertising revenue, A, and the monthly circulation, x, of a magazine are related approximately by the equation A = 6 √(x2 – 400), x ≥ 20, where A is given in thousands of dollars and x is measured in thousands of copies sold. At what rate

> Find the point(s) on the graph of y = (2x4 + 1)(x - 5) where the slope is 1.

> Suppose that the price p (in dollars) and the weekly demand, x (in thousands of units) of a commodity satisfy the demand equation 6p + x + xp = 94. How fast is the demand changing at a time when x = 4, p = 9, and the price is rising at the rate of $2 p

> Suppose that the price p (in dollars) and the weekly sales x (in thousands of units) of a certain commodity satisfy the demand equation 2p3 + x2 = 4500. Determine the rate at which sales are changing at a time when x = 50, p = 10, and the price is falli

> A point is moving along the graph of x3y2 = 200. When the point is at (2, 5), its x-coordinate is changing at the rate of -4 units per minute. How fast is the y-coordinate changing at that moment?

> A point is moving along the graph of x2 - 4y2 = 9. When the point is at (5, -2), its x-coordinate is increasing at the rate of 3 units per second. How fast is the y-coordinate changing at that moment?

> Suppose that x and y are both differentiable functions of t and are related by the given equation. Use implicit differentiation with respect to t to determine dy/dt in terms of x, y, and dx/dt. x2y2 = 2y3 + 1

> Suppose that x and y are both differentiable functions of t and are related by the given equation. Use implicit differentiation with respect to t to determine dy/dt in terms of x, y, and dx/dt. x2 + 2xy = y3

> Suppose that x and y are both differentiable functions of t and are related by the given equation. Use implicit differentiation with respect to t to determine dy/dt in terms of x, y, and dx/dt. y2 = 8 + xy

> Suppose that x and y are both differentiable functions of t and are related by the given equation. Use implicit differentiation with respect to t to determine dy/dt in terms of x, y, and dx/dt. 3xy - 3x2 = 4

> A 4500-year-old wooden chest was found in the tomb of the twenty-fifth century b.c. Chaldean king Meskalumdug of Ur. What percentage of the original 14C would you expect to find in the wooden chest?

> Suppose that x and y are both differentiable functions of t and are related by the given equation. Use implicit differentiation with respect to t to determine dy/dt in terms of x, y, and dx/dt. y4 - x2 = 1

> Suppose that x and y are both differentiable functions of t and are related by the given equation. Use implicit differentiation with respect to t to determine dy/dt in terms of x, y, and dx/dt. x4 + y4 = 1

> Find the point(s) on the graph of y = (x2 + 3x - 1)/x where the slope is 5.

> Suppose that x and y represent the amounts of two basic inputs for a production process and 10x1/2y1/2 = 600. Find dy/dx when x = 50 and y = 72.

> Suppose that x and y represent the amounts of two basic inputs for a production process and that the equation 30x1/3y2/3 = 1080 describes all input amounts where the output of the process is 1080 units. (a) Find dy/dx. (b) What is the marginal rate of

> The graph of x4 + 2x2y2 + y4 = 9x2 - 9y2 is a lemniscate similar to that in Fig. 6. (a) Find dy/dx by implicit differentiation. (b) Find the slope of the tangent line to the lemniscate at (√5, -1). Figure 6: У (x² + y²2 = 4x² - 4у

> The graph of x4 + 2x2y2 + y4 = 4x2 - 4y2 is the lemniscate in Fig. 6. (a) Find dy/dx by implicit differentiation. (b) Find the slope of the tangent line to the lemniscate at (√6/2, √2/2). Figure 6: У (x² + y²2 =

> Find the equation of the tangent line to the graph of x4 y2 = 144 at the point (2, 3) and at the point (2, -3).

> Find the equation of the tangent line to the graph of x2 y4 = 1 at the point (4, 12) and at the point (4, - 12 ).

> Use implicit differentiation of the equation to determine the slope of the graph at the given point. y2 = 3xy - 5; x = 2, y = 1

> According to legend, in the fifth century King Arthur and his knights sat at a huge round table. A round table alleged to have belonged to King Arthur was found at Winchester Castle in England. In 1976, carbon dating revealed the amount of radiocarbon in

> Use implicit differentiation of the equation to determine the slope of the graph at the given point. xy + y3 = 14; x = 3, y = 2

> Use implicit differentiation of the equation to determine the slope of the graph at the given point. √x + √y = 7; x = 9, y = 16

> Use implicit differentiation of the equation to determine the slope of the graph at the given point. xy3 = 2; x = - 14, y = -2

> The graph of y = (x2 - 1)4 (x2 + 1)5 is shown in Fig. 3. Find the coordinates of the local maxima and minima. Figure 3: -1 김 1 y = r2 - 1)(G2 + 1) 1

> Use implicit differentiation of the equation to determine the slope of the graph at the given point. y2 = x3 + 1; x = 2, y = -3

> Use implicit differentiation of the equation to determine the slope of the graph at the given point. 4y3 - x2 = -5; x = 3, y = 1

> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. x3y + xy3 = 4

> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. x2y + y2x = 3

> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. x2 + 4xy + 4y = 1

> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. x3 + y3 = x3y3

> In 1947, a cave with beautiful prehistoric wall paintings was discovered in Lascaux, France. Some charcoal found in the cave contained 20% of the 14C expected in living trees. How old are the Lascaux cave paintings? (Recall that the decay constant for 14

> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. (x + 1)2 (y - 1)2 = 1

> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. x3y2 - 4x2 = 1

> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. x2y3 = 6

> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. x(y + 2)5 = 8

> Find all x such that dy/dx = 0, where y = (x2 - 4)3 (2x2 + 5)5.

> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. xy3 = 2

> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. xy = 5

> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. x4 + 4y = x - 4y3

> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. 2x3 + y = 2y3 + x

> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. x3 + y3 = x2 + y2

> A sample of radioactive material has decay constant .25, where time is measured in hours. How fast will the sample be disintegrating when the sample size is 8 grams? For what sample size will the sample size be decreasing at the rate of 2 grams per day?

> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. x4 + (y + 3)4 = x2

> Compute f (g (x)), where f (x) and g (x) are the following: f (x) = x - 1, g (x) = 1/x + 1

> Compute f (g (x)), where f (x) and g (x) are the following: f (x) = x/x + 1, g (x) = x3

> One hundred shares of a technology stock were purchased on January 2, 1990, for $1200 and sold on January 2, 1998, for $12,500. What rate of interest compounded continuously did this investment earn?

> Find the inflection points on the graph of y = 1/(x2 + 1). (See Fig. 2.) Figure 2: -1 김 : f 2²+1 1

> Ten thousand dollars is invested at 6.5% interest compounded continuously. When will the investment be worth $41,787?

> One thousand dollars is deposited in a savings account at 6% yearly interest compounded continuously. How many years are required for the balance in the account to reach $2500?

> An investment earns 5.1% yearly interest compounded continuously and is currently growing at the rate of $765 per year. What is the current value of the investment?

> An investment earns 4.2% yearly interest compounded continuously. How fast is the investment growing when its value is $9000?

> A sample of radioactive material decays over time (measured in hours) with decay constant .2. The graph of the exponential function y = P(t) in Fig. 7 gives the number of grams remaining after t hours. Figure 7: (a) How much was remaining after 1 hour

> A population is growing exponentially with growth constant .04. In how many years will the current population double?

> Ten thousand dollars is deposited in a savings account at 4.6% yearly interest compounded continuously. (a) What differential equation is satisfied by A(t), the balance after t years? (b) What is the formula for A(t)? (c) How much money will be in the a

> An investment of $2000 yields payments of $1200 in 3 years, $800 in 4 years, and $500 in 5 years. Thereafter, the investment is worthless. What constant rate of return r would the investment need to produce to yield the payments specified? The number r i

> Verify that daily compounding is nearly the same as continuous compounding by graphing y = 100[1 + (.05/360)]360x, together with y = 100e0.05x in the window [0, 64] by [250, 2500]. The two graphs should appear the same on the screen. Approximately how fa

> Four thousand dollars is deposited in a savings account at 3.5% yearly interest compounded continuously. (a) What is the formula for A(t), the balance after t years? (b) What differential equation is satisfied by A(t), the balance after t years? (c) How

> Verify that lim m→∞ (1 + 1/m)m = e by taking m increasingly large and noticing that (1 + 1/m)m approaches 2.718.

> When $1000 is invested at r% interest (compounded continuously) for 10 years, the balance is f (r) dollars, where f is the function shown in Fig. 3. (a) What will the balance be at 7% interest? (b) For what interest rate will the balance be $3000? (c) If

> Find all x-coordinates of points (x, y) on the curve y = (x - 2)5 / (x - 4)3 where the tangent line is horizontal.

> The function A(t) in Fig. 2(a) gives the balance in a savings account after t years with interest compounded continuously. Figure 2(b) shows the derivative of A(t). (a) What is the balance after 20 years? (b) How fast is the balance increasing after 20 y

> The curve in Fig. 1 shows the growth of money in a savings account with interest compounded continuously. (a) What is the balance after 20 years? (b) At what rate is the money growing after 20 years? (c) Use the answers to parts (a) and (b) to determine

> A small amount of money is deposited in a savings account with interest compounded continuously. Let A(t) be the balance in the account after t years. Match each of the following answers with its corresponding question. Answers a. Pert b. A(3) c. A(0)

> Forty grams of a certain radioactive material disintegrates to 16 grams in 220 years. How much of this material is left after 300 years?

> Ten thousand dollars is deposited in a money market fund paying 8% interest compounded continuously. How much interest will be earned during the second year of the investment?

> Investment A is currently worth $70,200 and is growing at the rate of 13% per year compounded continuously. Investment B is currently worth $60,000 and is growing at the rate of 14% per year compounded continuously. After how many years will the two inve

> If the present value of $1000 to be received in 5 years is $559.90, what rate of interest, compounded continuously, was used to compute this present value?

> How much money must you invest now at 4.5% interest compounded continuously to have $10,000 at the end of 5 years?

> Find the present value of $2000 to be received in 10 years, if money may be invested at 8% with interest compounded continuously.

> Let A(t) be the balance in a savings account after t years, and suppose that A(t) satisfies the differential equation A’(t) = .045A(t), A(0) = 3000. (a) How much money was originally deposited in the account? (b) What interest rate is being earned? (c

> Find the present value of $1000 payable at the end of 3 years, if money may be invested at 8% with interest compounded continuously.

> Find the equation of the tangent line to the curve y = (x + 1) / (x - 1) at the point (2, 3).

> Differentiate the functions. y = (2x4 - x + 1)(-x5 + 1)

> A parcel of land bought in 1990 for $10,000 was worth $16,000 in 1995. If the land continues to appreciate at this rate, in what year will it be worth $45,000?

> In an animal hospital, 8 units of a sulfate were injected into a dog. After 50 minutes, only 4 units remained in the dog. Let f (t) be the amount of sulfate present after t minutes. At any time, the rate of change of f (t) is proportional to the value of

> A farm purchased in 2000 for $1 million was valued at $3 million in 2010. If the farm continues to appreciate at the same rate (with continuous compounding), when will it be worth $10 million?

> How is the account in Exercise 15 changing when the balance is 9,500 SFr?

> Suppose that the bank in Example 3 increased its fees by charging a negative annual interest rate of -.9%. Find the balance after two years in a savings account if P0 = 10, 000 SFr.

> If real estate in a certain city appreciates at the yearly rate of 15% compounded continuously, when will a building purchased in 2010 triple in value?

> If an investment triples in 15 years, what yearly interest rate (compounded continuously) does the investment earn?

> What yearly interest rate (compounded continuously) is earned by an investment that doubles in 10 years?

> How many years are required for an investment to double in value if it is appreciating at the yearly rate of 4% compounded continuously?

> Pablo Picasso’s Angel Fernandez de Soto was acquired in 1946 for a postwar splurge of $22,220. The painting was sold in 1995 for $29.1 million. What yearly rate of interest compounded continuously did this investment earn?

> Let A(t) = 5000e0.04t be the balance in a savings account after t years. (a) How much money was originally deposited? (b) What is the interest rate? (c) How much money will be in the account after 10 years? (d) What differential equation is satisfied by

> Find the equation of the tangent line to the curve y = (x - 2)5 (x + 1)2 at the point (3, 16).

> Ten grams of a radioactive material disintegrates to 3 grams in 5 years. What is the half-life of the radioactive material?

> Write each expression in the form ekx for a suitable constant k. (1/e3)2x, e1-x * e3x-1

> Write each expression in the form ekx for a suitable constant k. (e3)x/5, (1/e2)x

> Write each expression in the form ekx for a suitable constant k. (e2)x, (1/e)x

> Compute the given derivatives with the help of formulas (1)–(4). (a) d/dx (ex) |x=e (b) d/dx (ex) |x=1/e

> Calculate values of (10x – 1)/x for small values of x, and use them to estimate d/dx (10x) |x=0 . What is the formula for d/dx (10x)?

> Set Y1 = ex and use your calculator’s derivative command to specify Y2 as the derivative of Y1. Graph the two functions simultaneously in the window [-1, 3] by [-3, 20] and observe that the graphs overlap.

> (a) Graph y = ex. (b) Zoom in on the region near x = 0 until the curve appears as a straight line and estimate the slope of the line. This number is an estimate of d/dx ex at x = 0. Compare your answer with the actual slope, 1. (c) Repeat parts (a) and (