Differentiate the function. y = √(3x – 1) / x
> 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 (
> Find the equation of the tangent line to the graph of y = ex at x = 0. Then, graph the function and the tangent line together to confirm that your answer is correct.
> Sketch the graphs of the following functions. y = 2e-x
> Sketch the graphs of the following functions. y = -e-x + 1
> If dairy cows eat hay containing too much iodine 131, their milk will be unfit to drink. Iodine 131 has half-life of 8 days. If the hay contains 10 times the maximum allowable level of iodine 131, how many days should the hay be stored before it is fed t
> Sketch the graphs of the following functions. y = ex-1
> Compute the given derivatives with the help of formulas (1)–(4). (a) d/dx (ex) |x=1 (b) d/dx (ex) |x=-1
> Sketch the graphs of the following functions. y = ex/2
> Sketch the graphs of the following functions. y = 1 - ex
> Sketch the graphs of the following functions. y = e2x
> Compute the following derivatives. (a) Use the fact that e4x = (ex)4 to find d/dx (e4x). Simplify the derivative as much as possible. (b) Take an approach similar to the one in (a) and show that, if k is a constant, d/dx (ekx) = kekx.
> Compute the following derivatives. (a) d/dx (5ex) (b) d/dx (ex)10 (c) Use the fact that e2+x = e2 * ex to find d/dx (ex+2).
> Find the first and second derivatives. f (x) = ex/x
> Find the first and second derivatives. f (x) = ex(1 + x)2
> Differentiate the functions. y = (x + 3) / (2x + 1)2
> Find the equation of the tangent line to the curve y = ex/(x + ex) at (0, 1).
> Differentiate the function. y = √x + 2(2x + 1)2
> Find the equation of the tangent line to the curve y = ex/(1 + 2ex) at (0, 1/3).
> Find the slope of the tangent line to the curve y = xex at (1, e).
> Compute the given derivatives with the help of formulas (1)–(4). (a) d/dx (2x) |x=1/2 (b) d/dx (2x) |x=2
> Find the slope of the tangent line to the curve y = xex at (0, 0).
> Show that the tangent line to the graph of y = ex at the point (a, ea) is perpendicular to the tangent line to the graph of y = e-x at the point (a, e-a).
> Find the point on the graph of y = (1 + x2)ex where the tangent line is horizontal.
> Find the extreme points on the graph of y = x2ex, and decide which one is a maximum and which one is a minimum.
> The graph of y = x - ex has one extreme point. Find its coordinates and decide whether it is a maximum or a minimum.
> Radioactive cobalt 60 has a half-life of 5.3 years. Find its decay constant.
> Differentiate the following functions. y = 2ex + 1
> Differentiate the following functions. y = (ex – 1)/(ex + 1)
> Differentiate the function. y = √ [(x + 3)/(x2 + 1)]
> Differentiate the following functions. y = (x + 1)/ex
> Differentiate the following functions. y = ex/(x + 1)
> Differentiate the following functions. y = (1 + ex)(1 - ex)
> Compute the given derivatives with the help of formulas (1)–(4). (a) d/dx (2x) |x=1 (b) d/dx (2x) |x=-2
> Differentiate the following functions. y = 8ex(1 + 2ex)2
> Differentiate the following functions. y = (x2 + x + 1)ex
> Differentiate the following functions. y = xex
> Determine the growth constant k, then find all solutions of the given differential equation. y' = y/4
> Differentiate the following functions. y = (2x + 4 - 5ex)/4
> Differentiate the following functions. y = 3ex - 7x
> Find the slope–point form of the equation of the tangent line to the graph of ex at the point (a, ea).
> Differentiate the function. y = (x + 11 / x – 3)3
> Suppose that A = (a, b) is a point on the graph of ex. What is the slope of the graph of ex at the point A?
> Estimate the slope of ex at x = 0 by calculating the slope (eh – 1)/h of the secant line passing through the points (0, 1) and (h, eh). Take h = .01, .001, and .0001.
> Use the first and second derivative rules from Section 2.2 to show that the graph of y = ex has no relative extreme points and is always concave up.
> Find the point on the graph of f (x) = ex, where the tangent line is parallel to y = x.
> Show that d/dx (2.7x) |x=0 ≈ .99 by calculating the slope (2.7h – 1)/h of the secant line passing through the points (0, 1) and (h, 2.7h). Take h = .1, .01, and .001.
> Find an equation of the tangent line to the graph of f (x) = ex, where x = -1. (Use 1/e = .37.)
> The decay constant for the radioactive element cesium 137 is .023 when time is measured in years. Find its half-life.
> Solve each equation for x. 4ex(x2 + 1) = 0
> Solve each equation for x. ex(x2 - 1) = 0
> Solve each equation for x. e-x = 1
> Solve each equation for x. ex2-2x = e8
> Differentiate the function. y = 3/ (3√x + 1)
> Solve each equation for x. e1-x = e2
> Solve each equation for x. e5x = e20
> Write each expression in the form ekx for a suitable constant k. √(e-x*e7x), e-3x/e-4x
