Find the slope–point form of the equation of the tangent line to the graph of ex at the point (a, ea).
> 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
> Differentiate the function. y = √(3x – 1) / x
> 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
> 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
> Write each expression in the form ekx for a suitable constant k. (e4x*e6x)3/5, 1/e-2x
> Write each expression in the form ekx for a suitable constant k. (e5/e3)x, e4x+2*ex-2
> Ten grams of a radioactive substance with decay constant .04 is stored in a vault. Assume that time is measured in days, and let P(t) be the amount remaining at time t. (a) Give the formula for P(t) (b) Give the differential equation satisfied by P(t). (
> Show that d/dx (3x) |x=0 ≈ 1.1 by calculating the slope (3h – 1)/h of the secant line passing through the points (0, 1) and (h, 3h). Take h = .1, .01, and .001.
> In an expression of the form f (g (x)), f (x) is called the outer function and g (x) is called the inner function. Give a written description of the chain rule using the words inner and outer.
> Refer to Exercise 61. (a) What was the maximum value of the company during the first 6 months since it went public, and when was that maximum value attained? (b) Assuming that the value of one share will continue to increase at the rate that it did durin
> Refer to Exercise 61. (a) Find dx/dt |t=2.5 and dx/dt |t=4. Give an interpretation for these values. (b) Use the chain rule to find dW/dt |t=2.5 and dW/dt |t=4. Give an interpretation for these values. Exercise 61: After a computer software company went
> Refer to Exercise 61. Use the chain rule to find dW/dt |t=1.5 and dW/dt |t=3.5 . Give an interpretation for these values. Exercise 61: After a computer software company went public, the price of one share of its stock fluctuated according to the graph i
> Differentiate the function. y = 1 / (√x + 1)
> After a computer software company went public, the price of one share of its stock fluctuated according to the graph in Fig. 1(a). The total worth of the company depended on the value of one share and was estimated to be where x is the value of one sha
> Consider the functions of Exercise 59. Find d/dx g (f (x)) |x=1. Exercise 59: If f (x) and g (x) are differentiable functions, such that f (1) = 2, f ‘ (1) = 3, f ‘ (5) = 4, g(1) = 5, g ’ (1) = 6, g ‘ (2) = 7, and g ‘ (5) = 8, find d/dx f ( g (x)) |x=1.
> If f (x) and g (x) are differentiable functions, such that f (1) = 2, f ‘ (1) = 3, f ‘ (5) = 4, g(1) = 5, g ’ (1) = 6, g ‘ (2) = 7, and g ‘ (5) = 8, find d/dx f ( g (x)) |x=1.
> If f (x) and g (x) are differentiable functions, find g (x) if you know that f (x) = 1>x and d/dx f (g (x)) = (2x + 5)/(x2 + 5x – 4).
> A person is given an injection of 300 milligrams of penicillin at time t = 0. Let f (t) be the amount (in milligrams) of penicillin present in the person’s bloodstream t hours after the injection. Then, the amount of penicillin decays exponentially, and
> By trial and error, find a number of the form b = 2.(just one decimal place) with the property that the slope of the graph of y = bx at x = 0 is as close to 1 as possible.
> Determine the growth constant k, then find all solutions of the given differential equation. y' = 1.7y
> A population is growing exponentially with growth constant .05. In how many years will the current population triple?
> Determine the growth constant of a population that is growing at a rate proportional to its size, where the population triples in size every 10 years and time is measured in years.
> Determine the growth constant of a population that is growing at a rate proportional to its size, where the population doubles in size every 40 days and time is measured in days.
> The size of a certain insect population is given by P(t) = 300e0.01t, where t is measured in days. (a) How many insects were present initially? (b) Give a differential equation satisfied by P(t). (c) At what time will the initial population double? (d) A
> After t hours there are P(t) cells present in a culture, where P(t) = 5000e0.2t. (a) How many cells were present initially? (b) Give a differential equation satisfied by P(t). (c) When will the initial number of cells double? (d) When will 20,000 cells b
> Approximately 10,000 bacteria are placed in a culture. Let P(t) be the number of bacteria present in the culture after t hours, and suppose that P(t) satisfies the differential equation P (t) = .55P(t). (a) What is P(0)? (b) Find the formula for P(t). (c
> Let P(t) be the population (in millions) of a certain city t years after 2015, and suppose that P(t) satisfies the differential equation P (t) = .03P(t), P(0) = 4. (a) Use the differential equation to determine how fast the population is growing when it
> Differentiate the functions. y = (x2 – 1) / (x2 + 1)
> The initial size of a bacteria culture that grows exponentially was 10,000. After 1 day, there are 15,000 bacteria. (a) Find the growth constant if time is measured in days. (b) How long will it take for the culture to double in size?
> A colony of fruit flies exhibits exponential growth. Suppose that 500 fruit flies are present. Let P(t) denote the number of fruit flies t days later, and let k = .08 denote the growth constant. (a) Write a differential equation and initial condition tha
> A colony of fruit flies exhibits exponential growth. Suppose that 500 fruit flies are present. Let P(t) denote the number of fruit flies t days later, and let k = .08 denote the growth constant. (a) Write a differential equation and initial condition tha
> Determine the growth constant k, then find all solutions of the given differential equation. y' = .4y
> Let P(t) be the population (in millions) of a certain city t years after 2015, and suppose that P(t) satisfies the differential equation P (t) = .01P(t), P(0) = 2. (a) Find a formula for P(t). (b) What was the initial population, that is, the population
> Solve the given differential equation with initial condition. 5y = 3y’ , y(0) = 7
> Solve the given differential equation with initial condition. 6y’ = y, y(0) = 12
> Solve the given differential equation with initial condition. y’ -y/7 = 0, y(0) = 6
> Solve the given differential equation with initial condition. y’ - .6y = 0, y(0) = 5
> Solve the given differential equation with initial condition. y ’ = y, y(0) = 4
> Differentiate the functions. y = 1 / (x2 + x + 7)
