An investor initially invests $10,000 in a risky venture. Suppose that the investment earns 20% interest, compounded continuously, for 5 years and then 6% interest, compounded continuously, for 5 years thereafter. (a) How much does the $10,000 grow to after 10 years? (b) The investor has the alternative of an investment paying 14% interest compounded continuously. Which investment is superior over a 10-year period, and by how much?
> Suppose that P, y, and t are variables, where P is a function of y and y is a function of t. (a) Write the derivative symbols for the following quantities: (i) the rate of change of y with respect to t; (ii) the rate of change of P with respect to y;
> Many relations in biology are expressed by power functions, known as allometric equations, of the form y = kxa, where k and a are constants. For example, the weight of a male hognose snake is approximately 446x3 grams, where x is its length in meters. If
> The length, x, of the edge of a cube is increasing. (a) Write the chain rule for dV/dt, the time rate of change of the volume of the cube. (b) For what value of x is dV/dt equal to 12 times the rate of increase of x?
> Radium 226 is used in cancer radiotherapy. Let P(t) be the number of grams of radium 226 in a sample remaining after t years, and let P(t) satisfy the differential equation P (t) = -.00043P(t), P(0) = 12. (a) Find the formula for P(t). (b) What was the i
> The function f (x) = √(x2 - 6x + 10) has one relative minimum point for x ≥ 0. Find it.
> Find the x-coordinates of all points on the curve y = (-x2 + 4x - 3)3 with a horizontal tangent line.
> Find the equation of the line tangent to the graph of y = x/(√2 - x2) at the point (1, 1).
> Find the equation of the line tangent to the graph of y = 2x(x - 4)6 at the point (5, 10).
> Compute dy/dt |t=t0. y = √(x + 1), x = √(t + 1), t0 = 0
> Compute dy/dt |t=t0. y = (x + 1)/(x – 1), x = t2/4, t0 = 3
> Compute dy/dt |t=t0. y = (x2 - 2x + 4)2, x = 1/t + 1, t0 = 1
> Differentiate the function. y = [(3x2 + 2x + 2)(x - 2)]2
> Compute dy/dt |t=t0. y = x2 - 3x, x = t2 + 3, t0 = 0
> Compute dy/dx using the chain rule in formula (1). State your answer in terms of x only. y = (u2 + 2u)/(u + 1), u = x(x + 1) Chain Rule: The Chain Rule To differentiate f(g(x)), differentiate first the outside function f(x) and substitute g(x) for
> A sample of 8 grams of radioactive material is placed in a vault. Let P(t) be the amount remaining after t years, and let P(t) satisfy the differential equation P (t) = -.021P(t). (a) Find the formula for P(t) (b) What is P(0)? (c) What is the decay cons
> Compute dy/dx using the chain rule in formula (1). State your answer in terms of x only. y = u/2 + 2/u , u = x - x2 Chain Rule: The Chain Rule To differentiate f(g(x)), differentiate first the outside function f(x) and substitute g(x) for x in the
> Compute dy/dx using the chain rule in formula (1). State your answer in terms of x only. y = √(u + 1), u = 2x2 Chain Rule: The Chain Rule To differentiate f(g(x)), differentiate first the outside function f(x) and substitute g(x)
> Compute dy/dx using the chain rule in formula (1). State your answer in terms of x only. y = u3/2, u = 4x + 1 Chain Rule: The Chain Rule To differentiate f(g(x)), differentiate first the outside function f(x) and substitute g(x) for x in the result
> Compute d/dx f (g (x)), where f (x) and g (x) are: f (x) = x(x - 2)4, g (x) = x3
> Solve the following equations for x 22x – 4*2x - 32 = 0
> Solve the following equations for x 32x – 12*3x + 27 = 0
> Solve the following equations for x 22x+2 - 17*2x + 4 = 0
> Solve the following equations for x 22x - 6*2x + 8 = 0
> Solve the following equations for x 2x – 1/2x = 0
> Compute d/dx f (g (x)), where f (x) and g (x) are: f (x) = (x3 + 1)2, g (x) = x2 + 5
> Solve the following equations for x 2x – 8/22x = 0
> Solve the following equations for x (2 - 3x)5x + 4 * 5x = 0
> Write expression in the form 2kx or 3kx, for a suitable constant k. 82x/3, 93x/2, 16-3x/4
> Solve the following equations for x (1 + x)2-x - 5*2-x = 0
> Solve the following equations for x. 35x * 3x - 3 = 0
> Differentiate the functions. y = (5x + 1)(x2 - 1) + (2x + 1)/3
> Solve the following equations for x. 23x = 4 * 25x
> Solve the following equations for x. (32x * 32)4 = 3
> Solve the following equations for x. (2x+1 * 2-3)2 = 2
> Solve the following equations for x. 4(2.7)2x-1 = 10.8
> Compute d/dx f (g (x)), where f (x) and g (x) are: f (x) = 4/x + x2, g (x) = 1 - x4
> Solve the following equations for x. 3(2.7)5x = 8.1
> Solve the following equations for x. 24-x = 8
> Solve the following equations for x. 101-x = 100
> Solve the following equations for x. (3.2)x-3 = (3.2)5
> Write expression in the form 2kx or 3kx, for a suitable constant k. 27x, (3√2)x, (1/8)x
> Solve the following equations for x. (2.5)2x+1 = (2.5)5
> Differentiate the functions. y = [(-2x3 + x)(6x - 3)]4
> Differentiate the following functions. y = 5(√x - 1)4(√x - 2)2
> Differentiate the following functions. y = (2x + 1)5/2(4x - 1)3/2
> Differentiate the following functions. y = x(x5 - 1)3
> Compute d/dx f (g (x)), where f (x) and g (x) are: f (x) = x4 - x2, g (x) = x2 - 4
> Differentiate the following functions. y = 2(5 - x)3(6x - 1)
> Differentiate the following functions. y = (4x - 1)(3x + 1)4
> A stock portfolio increased in value from $100,000 to $117,000 in 2 years. What rate of interest, compounded continuously, did this investment earn?
> Solve the following equations for x. 10-x = 102
> From January 1, 2010, to January 1, 2017, the population of a state grew from 17 million to 19.3 million. (a) Give the formula for the population t years after 2010. (b) If this growth continues, how large will the population be in 2020? (c) In what year
> A piece of charcoal found at Stonehenge contained 63% of the level of 14C found in living trees. Approximately how old is the charcoal?
> The half-life of the radioactive element tritium is 12 years. Find its decay constant.
> One thousand dollars is deposited in a savings account at 10% interest compounded continuously. How many years are required for the balance in the account to reach $3000?
> Find the present value of $10,000 payable at the end of 5 years if money can be invested at 12% with interest compounded continuously.
> Compute d/dx f (g (x)), where f (x) and g (x) are: f (x) = 1/(1 + √x), g (x) = 1/x
> When a rod of molten steel with a temperature of 1800ËšF is placed in a large vat of water at temperature 60F, the temperature of the rod after t seconds is f (t) = 60(1 + 29e-0.15t)ËšF. The graph of this function is shown in Fig. 2.
> The growth of the yellow nutsedge weed is described by a logistic growth formula f (t) of type (9) in Section 5.4. A typical weed has length 8 centimeters after 9 days and length 48 centimeters after 25 days and reaches length 55 centimeters at maturity.
> Refer to Check Your Understanding 5.4. Out of 100 doctors in Group A, none knew about the drug at time t = 0, but 66 of them were familiar with the drug after 13 months. Find the formula for f (t). Check Your Understanding 5.4: A sociological study was
> Consider a demand function of the form q = ae-bp, where a and b are positive numbers. Find E(p), and show that the elasticity equals 1 when p = 1/b.
> A company can sell q = 1000p2e-0.02(p+5) calculators at a price of p dollars per calculator. The current price is $200. If the price is decreased, will the revenue increase or decrease?
> Solve the following equations for x. 52x = 52
> Find the percentage rate of change of the function f (p) = 1/(3p + 1) at p = 1.
> The herring gull population in North America has been doubling every 13 years since 1900. Give a differential equation satisfied by P(t), the population t years after 1900.
> For a certain demand function, E(8) = 1.5. If the price is increased to $8.16, estimate the percentage decrease in the quantity demanded. Will the revenue increase or decrease?
> Find E(p) for the demand function q = 4000 - 40p2, and determine if demand is elastic or inelastic at p = 5.
> Differentiate the function. y = (x + 1)3 / (x - 5)2
> Find the percentage rate of change of the function f (t) = 50e0.2t2 at t = 10.
> The current balance in a savings account is $1230, and the interest rate is 4.5%. At what rate is the compound amount currently growing?
> A few years after money is deposited into a bank, the compound amount is $1000, and it is growing at the rate of $60 per year. What interest rate (compounded continuously) is the money earning?
> You have 80 grams of a certain radioactive material, and the amount remaining after t years is given by the function f (t) shown in Fig. 1. (a) How much will remain after 5 years? (b) When will 10 grams remain? (c) What is the half-life of this radioacti
> The population of a certain country is growing exponentially. The total population (in millions) in t years is given by the function P(t). Match each of the following answers with its corresponding question. Answers a. Solve P(t) = 2 for t. b. P(2) c. P
> A colony of bacteria is growing exponentially with growth constant .4, with time measured in hours. Determine the size of the colony when the colony I growing at the rate of 200,000 bacteria per hour. Determine the rate at which the colony will be growin
> Find b so that 8-x/3 = bx for all x.
> The population of a city t years after 1990 satisfies the differential equation y = .02y. What is the growth constant? How fast will the population be growing when the population reaches 3 million people? At what level of population will the population
> Two different bacteria colonies are growing near a pool of stagnant water. The first colony initially has 1000 bacteria and doubles every 21 minutes. The second colony has 710,000 bacteria and doubles every 33 minutes. How much time will elapse before th
> The atmospheric pressure P(x) (measured in inches of mercury) at height x miles above sea level satisfies the differential equation P (x) = -.2P(x). Find the formula for P(x) if the atmospheric pressure at sea level is 29.92.
> Differentiate the functions. y = (-x3 + 2) (x/2 – 1)
> The population (in millions) of a state t years after 2010 is given by the graph of the exponential function y = P(t) with growth constant .025 in Fig. 6. [In parts (c) and (d) use the differential equation satisfied by P(t).] Figure 6: (a) What is the
> Simplify the following. (ex2)3
> Calculate the following. 40.2 * 40.3
> Calculate the following. 95/2 / 93/2
> Calculate the following. 81/2 * 21/2
> Differentiate the function. y = x / ln x
> Differentiate the function. y = ln(x6 + 3x4 + 1)
> Solve the following equations for t. 2e- 0.3t = 1
> Find a number b such that the function f (x) = 3-2x can be written in the form f (x) = bx.
> Calculate the following. (25/7)14/5
> Solve the following equations for t. 2 ln t = 5
> Compute d/dx f (g (x)), where f (x) and g (x) are: f (x) = 1/x , g (x) = 1 - x2
> Solve the following equations for t. 3et/2 - 12 = 0
> Solve the following equations for t. 3e2t = 15
> Solve the following equations for t. ln( ln 3t) = 0
> Solve the following equations for t. t ln t = e
> Simplify the following expressions. [e ln x]2
> Simplify the following expressions. e-5 ln 1
> Simplify the following expressions. e2 ln 2
> Simplify the following expressions. ln x2 / ln x3
> Write expression in the form 2kx or 3kx, for a suitable constant k. (3-x * 3x/5)5, (161/4 * 16-3/4)3x
