Section 4.1
1) Evaluate each of the following expressions. (1/4)^3, (1/27)^(1/3), 8^(2/3), (3/7)^2, 100^(5/2), 25^(3/2)
2) Using the properties of exponents simplify the following expressions. (3^2)/(3^5), (1/6)^(-2), (32^(1/2))*(2^(1/2)), (8^(5/2)) * (1/2)^(5/2)
3) Using the properties of exponents simplify the following expressions. (3^3)*(3^2), (1/5)^3*(5^3), (2^8)^(1/4), [(8^-2)(8^(5/3))]^3
4) Let f(x) = 4^(x+1). Evaluate f(-4), f(-1/2), f(2), f(-5/2).
5) Let g(x) = 1.175^x. Evaluate g(1.3), g(170), g(40), g(12.5).
6) Solve the equation 4^(x+1) = 64 for x.
7) Solve the equation 4^2 = (x+2)^2 for x.
8) Sketch and describe the graph of 2^(-x/2).
9) Sketch and describe the graph of f(x) = (1/3)^x
10) Sketch and describe the graph of f(x) = 4^(-x).
11) Sketch and describe the graph of f(x) = 4^|x| (|x| denotes the absolute value of x.)
12) Suppose that the annual rate of inflation averages over 6% per year over the next 10 years. Given this, the approximate cost of goods is given by C(t) = P(1.06)^t, 0 <= t <=10 where t it time in years and P is the present cost. If the price of a movie ticket is currently $7.95 estimate the cost of the ticket in 10 years.
13) After t years, the initial mass of 20 grams of a radioactive element whose half-life is 35 years is given by y = 20((1/2)^(t/35)), t >= 0. How much of the initial mass remains after 60 years? Using a graphing calculator or something similar, find the amount of time required for the mass to decay to only 1 gram
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Section 4.2
1) Using the properties of exponents simplify the following expressions. (1/e)^-3, (e^2/e^5)^-1, e^2/e^4, 1/(e^-2)
2) 4.2.4 Using the properties of exponents simplify the following expressions. (e^-5)^(3/5), (e^3)/(e^(-1/3)), (e^-1)^-3, (e^4)(e^-3/4)
3) Solve the equation e^(x+1) = 1 for x.
4) 4.2.8 Solve the equation e^(-1/2x) = sqrt(e) for x. (sqrt(e) is the square root of e or e^1/2)
5) Sketch and describe the graph of the function f(x) = e^(-2x).
6) 4.2.20 Sketch and describe the graph of the function f(x) = e^(x/2)
7) 4.2.28 Use a graphing calculator or something similar to graph the function f(x) = (1/2)(e^x - e^(-x)). Determine whether the function has any horizontal asymptotes and discuss the continuity of the function. Explain how you could sketch the graph 'by hand', without the use of the calculator.
8) 4.2.32 Find the balance A of $2500 invested at 5% for 40 years when the interest is compounded 1, 2, 4, 12, and 365 times a year, and also when it is compounded continuously.
9) How much should be deposited into an account paying 8.1% interest compounded monthly in order to have a balance of $17,201.66 three years from now?
10) If your typing rate at clock time t is N = 95 / (1 + 8.5 e^(-.12 t)) words / minute, where t is clock time in weeks, then
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Section 4.3
1) Find the slope of the tangent line of the function y = e^(-3x) at the point (0,1)
2) 4.3.8 Find the derivative of f(x) = e^(1/x).
3) Find the derivative of g(x) = e^sqrt(x+3).
4) 4.3.12 Find the derivative of y = 5(x^2)(e^-x).
5) Find the derivative of y = (2x^2)(e^x) - (4x)(e^x) + 4e^x.
6) 4.3.16 Find the derivative of y = (e^-x + e^x)^3
7) Find dy/dx implicitly for the equation (x^3)y + xe^x - 12 = 3
8) 4.3.26 Find dy/dx implicitly for the equation e^(xy) + x^2- y^2 = 0.
9) Graph the function f(x) = 1/2(e^x + e^(-x)) and give its extrema, points of inflection, and asymptotes.
10) Give the equation of tangent line to the graph of e^(4x-2)^2 at the point (0, 1).
11) Find the extrema of the function y = x^2 e^(-x).
12) 4.3.42 The Ebbinghaus Model for human memory is p = (100-a)e^(-bt) + a where p is percent retained after t weeks. If a = 20 and b = 0.5 at what rate is information being retained after 1 week and after 3 weeks? At what rate is memory being lost at t = 3 weeks?
11) *4.3.44 From 1995 to 2002, the numbers y (in millions) of employed people in the United States can be modeled by y = 115.46 + 1.592t + 0.0552t^2 - 0.00004e^t where t = 5 corresponds to the year 1995. Find the rates of change of the number of employed people in 1996, 1998, 2001.
12) *4.4.46 A survey of certain group of college students has determined that the mean height of females is 65 inches with a standard deviation of 2.9 inches. Assuming that the data can be modeled with the normal distribution, find a model for the data. Find the derivative of the model and show that f ' > 0 when x < mu and f ' < 0 when x > mu.
13) Using a graphing calculator or something similar, graph the normal probability density function with sigma = 1 and mu = -3, -1, 1, 2 in the same window.
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Section 4.4
1) 4.4.2 Write ln .056 = 2.8824 as an exponential equation. x
2) 4.4.4 Write ln 0.047 = -3.0576.. as an exponential equation.
3) 4.4.6 Write e^3 = 20.086... as a logarithmic equation.
4) 4.4.8 Write e^0.25 = 1.2840 as a logarithmic equation.
5) *4.4.10 Sketch and describe the function f(x) = 5 + ln(x). x
6) *4.4.12 Sketch and describe the function f(x) = -ln(x + 3).
7) *4.4.16 Sketch and describe the function f(x) = ln |x|.
8) *4.4.22 Analytically show that f(x) = e^(x/3) and g(x) = ln (x^3) are inverse functions.
9) 4.4.28 Simplify the expression 1/3 [ 2 ln(x+3) + ln x - ln(x^2-1) ] x
10) *4.4.30 Without using a calculator and given ln 2 ~= .7 and ln 3 ~= 1.1 approximate ln(.5), ln(36), ln(6^(1/4)) and ln(1/108). (~= here means "is about equal to").
11) 4.4.32 Write the expression, ln (2/7), as a sum, difference or product of logarithms.
12) 4.4.34 Write the expression, ln (x/yz), as a sum, difference or product of logarithms.
13) 4.4.36 Write the expression, ln (sqrt(x^2/(x-3))), as a sum, difference or product of logarithms.
14) 4.4.38 Write the expression, ln (x*(x^3 + 1)^(1/4)), as a sum, difference or product of logarithms.
15) 4.4.40 Write the expression, ln (4x/sqrt(x^2 -4)), as a sum, difference or product of logarithms.
16) 4.4.42 Write ln(3x+3) + ln(3x-2) as a single logarithm.
17) 4.4.44 Write 2ln(3) - (1/3)ln(x^3 +8) as a single logarithm.
18) 4.4.46 Write (1/2)[2ln(x-3) + ln x^2 - ln(x^2-4x+4)] as a single logarithm.
19) 4.4.48 Write 3[ln (x+2) + (1/3)ln(x+2)] as a single logarithm.
20) 4.4.50 Write (1/3)ln(x-5) + (4/3)ln(x+5) as a single logarithm.
21) 4.4.56 Solve 400 e^(-.0174 t) = 1000 for x. x
22) 4.4.58 Solve 250e^(-0.0274t) = 2000 for t.
23) 4.4.64 Solve 3^(1-2x) = 10 for x.
24) 4.4.68 Solve 2500(1+0.07/12)^12t = 15000 for t.
25) 4.4.72 The demand function p = p = 250 - .8 e^(.005x) relates the price of an object p, versus the number of units sold x. Find the number of units sold when the price is $200 and $125. x
26) 4.4.76 Carbon Dating uses the fact that Carbon-14 in an organism decays once the organism has died with a half-life of 5715 years. The ratio of Carbon-14 atoms to isotopes of Carbon is given by the equation R = (10^-12)(1/2)^(t/5715). Where t is time in years and t = 0 represents the time that the object dated died. Find t when R = 0.23 ^ 10^-12.
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Section 4.5
1) 4.5.2 Find the slope of the tangent line to the function y = ln (x^(3/2)) at the point (1,0).
2) 4.5.4 Find the slope of the tangent line to the function y = ln(x^(1/3)) at the point (1,0).
3) 4.5.6 Find the derivative of f(x) = ln 3x.
4) 4.5.8 Find the derivative of f(x) = ln(3-x^4).
5) 4.5.10 Find the derivative of f(x) = ln((1-x)^(1/3)). x
6) 4.5.12 Find the derivative of y = (ln x^3)^3.
7) 4.5.16 Find the derivative of y = ln ((x+1)/(x^2+1)).
8) 4.5.22 Find the derivative of y = ln(3x*sqrt(9+x^2))
9) 4.5.25 Find the derivative of g(x) = ln( (e^x + e^-x) / 2) x
10) 4.5.28 Write the expression 4^x with base e.
11) 4.5.30 Write the expression log{base 3}(x) with base e.
12) 4.5.32 Using a calculator evaluate log{base 5}(12) to the nearest thousandth.
13) 4.5.34 Using a calculator evaluate log{base 7}(2/9) to the nearest thousandth.
14) 4.5.36 Using a calculator evaluate log{base 2/3}(32) to the nearest thousandth.
15) 4.5.38 Find the derivative of y = (1/3)^x.
16) 4.5.40 Find the derivative of g(x) = log{base 7}(x).
17) 4.5.42 Find the derivative of y = 5^(3x).
18) 4.5.44 Find the derivative of f(x) = 7^(x^3).
19) 4.5.46 Find the derivative of y = x(5^(x+2)).
20) 4.5.48 Determine the equation of the tangent line to the function y = (ln x) / x at the point (e^2, 2/(e^2)).
21) 4.5.50 Determine the equation of the tangent line to the function g(x) = log{base 3}(2x-1) at the point (5,2).
22) Determine the equation of tangent line to the function y = 25^(2x^2) at the point (-1/2,5).
23) 4.5.56 Find the second derivative of the function f(x) = 5 + 3ln x.
24) 4.5.58 Find the second derivative of the function log{base 10}(x).
25) 4.5.59 The relationship between the number of decibels, dB , and the intensity of a sound I in watts per cm^2 is given by dB = 10 log{base 10}(I/10^-16). Find the rate of change in the number of decibels, with respect to intensity, when the intensity is 10^-4 watts/cm^2. x
26) 4.5.60 The temperature T (in Fahrenheit) at which water boils at various observed pressures p (pounds per square inch) can be modeled with good accuracy by T = 87.97 + 34.96 ln p + 7.91sqrt(p). Find the rate of change of the temperature with respect to pressure.
27) 4.5.64 Find the equation of the tangent line to the function f(x) = ln(x(sqrt(x+3)) at the point (1.2,0.9).
28) 4.5.68 Using the first and second derivative tests, analyze the function y = (1/3)x^2 + ln x.
29) 4.5.72 Using the first and second derivative tests, analyze the function y = (ln x)^3.
30) 4.5.76 Find dx/dp where x = 250 - 25ln(ln p) is the demand function. Interpret dx/dp when the price is $10.
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Section 4.6
1) 4.6.1. Find the exponential function y = Ce^(kt) which passes through the points (3,1/2) and (4,5). x
2) 4.6.10 Solve dy/dt = 5.2y for y when y(0) = 18.
3) 4.6.14 How much is left of an initial amount of Carbon-14(Half-life of 5715 years) of 4 grams after 1000 and 10000 years?
4) 4.6.18 Find the half-life of a radioactive material if after 2 years 99.32% of the initial amount remains.
5) 4.6.25 If an initial investment of $1000 is continuously compounded at 12% how long does it take the investment to double? What is the amount after 10 and 25 years? x
6) 4.6.26 If an initial investment of $25000 is continuously compounded at 9.5% how long does it take the investment to double? What is the amount after 10 and 25 years?
7) 4.6.42 Management for a certain company requires that a new employee be producing 20 units per day after 30 days on the job. The learning curve model for this company is N = 30(1- e^kt). Find k such that this minimal requirement is met and then find out how long would it take someone who fits this model to be producing 25 units a day.
8) 4.6.44 The demand function for a certain product can be modeled as p = Ce^(kx). When p = $5 and x = 1000 units and p = $4 and x = 400 units solve for C and k.