CALCULUS+ITS APPLICATIONS-MYMATHLAB
15th Edition
ISBN: 9780137590438
Author: Goldstein
Publisher: PEARSON
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Textbook Question
Chapter 4.3, Problem 23E
In Exercises 21-26, simplify the function before
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A ladder 25 feet long is leaning against the wall of a building. Initially, the foot of the ladder is 7 feet from the wall. The foot of the ladder begins to slide at a rate of 2 ft/sec, causing the top of the ladder to slide down the wall. The location of the foot of the ladder, its x coordinate, at time t seconds is given by
x(t)=7+2t.
wall
y(1)
25 ft. ladder
x(1)
ground
(a) Find the formula for the location of the top of the ladder, the y coordinate, as a function of time t. The formula for y(t)= √ 25² - (7+2t)²
(b) The domain of t values for y(t) ranges from 0
(c) Calculate the average velocity of the top of the ladder on each of these time intervals (correct to three decimal places):
. (Put your cursor in the box, click and a palette will come up to help you enter your symbolic answer.)
time interval
ave velocity
[0,2]
-0.766
[6,8]
-3.225
time interval
ave velocity
-1.224
-9.798
[2,4]
[8,9]
(d) Find a time interval [a,9] so that the average velocity of the top of the ladder on this…
Total marks 15
3.
(i)
Let FRN Rm be a mapping and x = RN is a given
point. Which of the following statements are true? Construct counterex-
amples for any that are false.
(a)
If F is continuous at x then F is differentiable at x.
(b)
If F is differentiable at x then F is continuous at x.
If F is differentiable at x then F has all 1st order partial
(c)
derivatives at x.
(d) If all 1st order partial derivatives of F exist and are con-
tinuous on RN then F is differentiable at x.
[5 Marks]
(ii) Let mappings
F= (F1, F2) R³ → R² and
G=(G1, G2) R² → R²
:
be defined by
F₁ (x1, x2, x3) = x1 + x²,
G1(1, 2) = 31,
F2(x1, x2, x3) = x² + x3,
G2(1, 2)=sin(1+ y2).
By using the chain rule, calculate the Jacobian matrix of the mapping
GoF R3 R²,
i.e., JGoF(x1, x2, x3). What is JGOF(0, 0, 0)?
(iii)
[7 Marks]
Give reasons why the mapping Go F is differentiable at
(0, 0, 0) R³ and determine the derivative matrix D(GF)(0, 0, 0).
[3 Marks]
5.
(i)
Let f R2 R be defined by
f(x1, x2) = x² - 4x1x2 + 2x3.
Find all local minima of f on R².
(ii)
[10 Marks]
Give an example of a function f: R2 R which is not bounded
above and has exactly one critical point, which is a minimum. Justify briefly
Total marks 15
your answer.
[5 Marks]
Chapter 4 Solutions
CALCULUS+ITS APPLICATIONS-MYMATHLAB
Ch. 4.1 - Can a function such as f(x)=53x be written in the...Ch. 4.1 - Solve the equation 7263x=28.Ch. 4.1 - Prob. 1ECh. 4.1 - Prob. 2ECh. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Prob. 7ECh. 4.1 - Write each expression in Exercises 1-14 in the...
Ch. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Prob. 10ECh. 4.1 - Prob. 11ECh. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Prob. 13ECh. 4.1 - Prob. 14ECh. 4.1 - Find a number b such that the function f(x)=32x...Ch. 4.1 - Find b so that 8x/3=bx for all x.Ch. 4.1 - Solve the following equations for x. 52x=52Ch. 4.1 - Solve the following equations for x. 10x=102Ch. 4.1 - Solve the following equations for x....Ch. 4.1 - Solve the following equations for x....Ch. 4.1 - Solve the following equations for x. 101x=100Ch. 4.1 - Solve the following equations for x. 24x=8Ch. 4.1 - Solve the following equations for x. 3(2.7)5x=8.1Ch. 4.1 - Solve the following equations for x....Ch. 4.1 - Solve the following equations for x. (2x+123)2=2Ch. 4.1 - Solve the following equations for x. (32x32)4=3Ch. 4.1 - Solve the following equations for x. 23x=425xCh. 4.1 - Solve the following equations for x. 35x3x3=0Ch. 4.1 - Solve the following equations for x. (1+x)2x52x=0Ch. 4.1 - Prob. 30ECh. 4.1 - Solve the following equations for x. 2x822x=0Ch. 4.1 - Prob. 32ECh. 4.1 - Solve the following equations for x. [Hint: In...Ch. 4.1 - Prob. 34ECh. 4.1 - Solve the following equations for x. [Hint: In...Ch. 4.1 - Prob. 36ECh. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - Prob. 42ECh. 4.1 - Prob. 43ECh. 4.1 - Prob. 44ECh. 4.1 - Prob. 45ECh. 4.2 - Solve the following equation for x: e6x=e3.Ch. 4.2 - Differentiate y=(x+ex)4Ch. 4.2 - Show that ddx(3x)|x=01.1 by calculating the slope...Ch. 4.2 - Show that ddx(2.7x)|x=0.99 by calculating the...Ch. 4.2 - In Exercises 3-6, compute the given derivatives...Ch. 4.2 - Prob. 4ECh. 4.2 - Prob. 5ECh. 4.2 - Prob. 6ECh. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Prob. 12ECh. 4.2 - Solve each equation for x. e5x=e20Ch. 4.2 - Prob. 14ECh. 4.2 - Solve each equation for x. ex22x=e8Ch. 4.2 - Prob. 16ECh. 4.2 - Solve each equation for x. ex(x21)=0Ch. 4.2 - Solve each equation for x. 4ex(x2+1)=0Ch. 4.2 - Find an equation of the tangent line to the graph...Ch. 4.2 - Prob. 20ECh. 4.2 - Use the first and second derivative rules from...Ch. 4.2 - Prob. 22ECh. 4.2 - Suppose that A=(a,b) is a point on the graph of...Ch. 4.2 - Find the slope-point form of the equation of the...Ch. 4.2 - Differentiate the following functions. y=3ex7xCh. 4.2 - Differentiate the following functions. y=2x+45ex4Ch. 4.2 - Differentiate the following functions. y=xexCh. 4.2 - Differentiate the following functions....Ch. 4.2 - Differentiate the following functions....Ch. 4.2 - Differentiate the following functions....Ch. 4.2 - Differentiate the following functions. y=exx+1Ch. 4.2 - Prob. 32ECh. 4.2 - Differentiate the following functions. y=ex1ex+1Ch. 4.2 - Differentiate the following functions. y=ex+1Ch. 4.2 - The graph of y=xex has one extreme point. Find its...Ch. 4.2 - Prob. 36ECh. 4.2 - Find the point on the graph of y=(1+x2)ex where...Ch. 4.2 - Prob. 38ECh. 4.2 - Find the slope of the tangent line to the curve...Ch. 4.2 - Find the slope of the tangent line to the curve...Ch. 4.2 - Find the equation of the tangent line to the curve...Ch. 4.2 - Find the equation of the tangent line to the curve...Ch. 4.2 - Find the first and second derivatives....Ch. 4.2 - Find the first and second derivatives. f(x)=exxCh. 4.2 - Compute the following derivatives. ddx(5ex)...Ch. 4.2 - Prob. 46ECh. 4.2 - Prob. 47ECh. 4.2 - Prob. 48ECh. 4.2 - Prob. 49ECh. 4.2 - Prob. 50ECh. 4.2 - Prob. 51ECh. 4.2 - Prob. 52ECh. 4.2 - Prob. 53ECh. 4.2 - Prob. 54ECh. 4.2 - Prob. 55ECh. 4.2 - Prob. 56ECh. 4.3 - Differentiate tet2Ch. 4.3 - Differentiate [ e3x(1+e6x) ]12.Ch. 4.3 - Differentiate the following functions. f(x)=e2x+3Ch. 4.3 - Differentiate the following functions. f(x)=e3x2Ch. 4.3 - Differentiate the following functions. f(x)=e4x2xCh. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions. f(x)=eexCh. 4.3 - Differentiate the following functions. f(x)=e1xCh. 4.3 - Differentiate the following functions. f(x)=exCh. 4.3 - Differentiate the following functions. f(x)=ex2+1Ch. 4.3 - Differentiate the following functions. f(x)=7ex7Ch. 4.3 - Differentiate the following functions. f(x)=10ex25Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions. f(x)=eeexCh. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions. f(x)=ex+1Ch. 4.3 - Differentiate the following functions. f(x)=eexCh. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - An Investment Portfolio The value of an investment...Ch. 4.3 - Depreciation of Assets The value of the computer t...Ch. 4.3 - The Most Expensive Artwork to Date The highest...Ch. 4.3 - Appreciation of Assets A painting purchased in...Ch. 4.3 - Velocity and Acceleration The velocity of the...Ch. 4.3 - Velocity and Acceleration Suppose the velocity of...Ch. 4.3 - Heights of a Plant The height of a certain plant,...Ch. 4.3 - Heights of a Plant The length of a certain weed,...Ch. 4.3 - Gompertz Growth Curve Let aandb be positive...Ch. 4.3 - Find dydx if y=e(110)ex2.Ch. 4.3 - Size of Tumor In a study, a cancerous tumor was...Ch. 4.3 - Height of a Plant Let f(t) be the function from...Ch. 4.4 - Find lne.Ch. 4.4 - Solve e3x=2 using the natural logarithm function.Ch. 4.4 - Find ln(e).Ch. 4.4 - Find ln(1e2).Ch. 4.4 - If ex=5, Write x in terms of the natural...Ch. 4.4 - If ex=3.2, Write x in terms of the natural...Ch. 4.4 - If lnx=1, Write x using the exponential function.Ch. 4.4 - If lnx=4.5, Write x using the exponential...Ch. 4.4 - Simplify the following expression. lne3Ch. 4.4 - Simplify the following expression. eln4.1Ch. 4.4 - Simplify the following expression. eeln1Ch. 4.4 - Simplify the following expression. ln(e2lne)Ch. 4.4 - Simplify the following expression. ln(lne)Ch. 4.4 - Simplify the following expression. e4ln1Ch. 4.4 - Simplify the following expression. e2lnxCh. 4.4 - Simplify the following expression. exln2Ch. 4.4 - Simplify the following expression. e2ln7Ch. 4.4 - Simplify the following expression. e2ln7Ch. 4.4 - Simplify the following expression. elnx+ln2Ch. 4.4 - Simplify the following expression. eln32lnxCh. 4.4 - Solve the following equations for x. e2x=5Ch. 4.4 - Solve the following equations for x. e13x=4Ch. 4.4 - Solve the following equations for x. ln(4x)=12Ch. 4.4 - Prob. 22ECh. 4.4 - Solve the following equations for x. lnx2=9Ch. 4.4 - Prob. 24ECh. 4.4 - Solve the following equations for x. 6e0.00012x=3Ch. 4.4 - Prob. 26ECh. 4.4 - Solve the following equations for x. ln3x=ln5Ch. 4.4 - Prob. 28ECh. 4.4 - Solve the following equations for x. ln(ln3x)=0Ch. 4.4 - Prob. 30ECh. 4.4 - Solve the following equations for x. 2ex/39=0Ch. 4.4 - Prob. 32ECh. 4.4 - Prob. 33ECh. 4.4 - Prob. 34ECh. 4.4 - Prob. 35ECh. 4.4 - Prob. 36ECh. 4.4 - Solve the following equations for x. 4exe2x=6Ch. 4.4 - Prob. 38ECh. 4.4 - The graph of f(x)=5x+ex is shown in fig. 4. Find...Ch. 4.4 - Prob. 40ECh. 4.4 - Prob. 41ECh. 4.4 - Prob. 42ECh. 4.4 - Prob. 43ECh. 4.4 - Find the x-intercept of y=(x1)2ln(x+1),x1.Ch. 4.4 - In Exercise 45- 46, find the coordinates of each...Ch. 4.4 - In Exercise 45- 46, find the coordinates of each...Ch. 4.4 - Solve for t. e0.05t4e0.06t=0Ch. 4.4 - Solve for t. 4e0.01t3e0.04t=0Ch. 4.4 - Prob. 49ECh. 4.4 - Wind Velocity Under certain geographic conditions,...Ch. 4.4 - Prob. 51ECh. 4.4 - Prob. 52ECh. 4.4 - Prob. 53ECh. 4.4 - Prob. 54ECh. 4.4 - Prob. 55ECh. 4.5 - Differentiate f(x)=1ln(x4+5).Ch. 4.5 - Differentiate f(x)=ln(lnx).Ch. 4.5 - Prob. 3CYUCh. 4.5 - Differentiate the following functions. y=3lnx+ln2Ch. 4.5 - Differentiate the following functions. y=lnxln3Ch. 4.5 - Differentiate the following functions. y=x2lnx2Ch. 4.5 - Differentiate the following functions. y=3lnxxCh. 4.5 - Differentiate the following functions. y=exlnxCh. 4.5 - Differentiate the following functions. y=e1+lnxCh. 4.5 - Differentiate the following functions. y=lnxxCh. 4.5 - Prob. 8ECh. 4.5 - Differentiate the following functions. y=lnx2Ch. 4.5 - Prob. 10ECh. 4.5 - Differentiate the following functions. y=ln(1x)Ch. 4.5 - Prob. 12ECh. 4.5 - Differentiate the following functions. y=ln(3x4x2)Ch. 4.5 - Prob. 14ECh. 4.5 - Differentiate the following functions. y=1lnxCh. 4.5 - Differentiate the following functions. y=lnxln2xCh. 4.5 - Differentiate the following functions. y=lnxln2xCh. 4.5 - Differentiate the following functions. y=(lnx)2Ch. 4.5 - Differentiate the following functions....Ch. 4.5 - Differentiate the following functions....Ch. 4.5 - Find the second derivatives. d2dt2(t2lnt)Ch. 4.5 - Find the second derivatives. d2dt2ln(lnt)Ch. 4.5 - The graph of f(x)=(lnx)/x is shown in Fig.4. Find...Ch. 4.5 - The graph of f(x)=x/(lnx+x) is shown in Fig.5....Ch. 4.5 - Write the equation of the tangent line to the...Ch. 4.5 - The function f(x)=(lnx+1)/x has a relative extreme...Ch. 4.5 - Determine the domain of definition of the given...Ch. 4.5 - Find the equations of the tangent lines to the...Ch. 4.5 - Find the coordinates of the relative extreme point...Ch. 4.5 - Repeat the previous exercise with y=xlnx.Ch. 4.5 - The graphs of y=x+lnx and y=ln2x are shown in...Ch. 4.5 - Prob. 32ECh. 4.5 - Prob. 33ECh. 4.5 - The function y=2x2ln4x (x0) has one minimum point....Ch. 4.5 - A Demand Equation If the demand equation for a...Ch. 4.5 - Total Revenue Suppose that the total revenue...Ch. 4.5 - An Area ProblemFind the maximum area of a...Ch. 4.5 - Analysis of the Effectiveness of an Insect...Ch. 4.6 - Differentiate f(x)=ln[ exx(x+1)6 ].Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Simplify the following expressions. ln5+lnxCh. 4.6 - Simplify the following expressions. lnx5lnx3Ch. 4.6 - Simplify the following expressions. 12ln9Ch. 4.6 - Simplify the following expressions. 3ln12+ln16Ch. 4.6 - Simplify the following expressions. ln4+ln6ln12Ch. 4.6 - Simplify the following expressions. ln2lnx+ln3Ch. 4.6 - Simplify the following expressions. e2lnxCh. 4.6 - Simplify the following expressions. 32ln45ln2Ch. 4.6 - Simplify the following expressions. 5lnx12lny+3lnzCh. 4.6 - Simplify the following expressions. elnx2+3lnyCh. 4.6 - Simplify the following expressions. lnxlnx2+lnx4Ch. 4.6 - Prob. 12ECh. 4.6 - Simplify the following expressions. Which is...Ch. 4.6 - Simplify the following expressions. Which is...Ch. 4.6 - Evaluate the given expressions. Use ln2=.69 and...Ch. 4.6 - Evaluate the given expressions. Use ln2=.69 and...Ch. 4.6 - Evaluate the given expressions. Use ln2=.69 and...Ch. 4.6 - Prob. 18ECh. 4.6 - Which of the following is the same as 4ln2x? a....Ch. 4.6 - Prob. 20ECh. 4.6 - Which of the following is the same as ln8x2ln2x?...Ch. 4.6 - Which of the following is the same as ln9x2? a....Ch. 4.6 - Solve the given equation for x. lnxlnx2+ln3=0Ch. 4.6 - Solve the given equation for x. lnx2ln3=0Ch. 4.6 - Solve the given equation for x. lnx42lnx=1Ch. 4.6 - Solve the given equation for x. lnx2ln2x+1=0Ch. 4.6 - Solve the given equation for x. (lnx)21=0Ch. 4.6 - Solve the given equation for x. 3lnxln3x=0Ch. 4.6 - Solve the given equation for x. lnx=lnxCh. 4.6 - Solve the given equation for x. 2(lnx)2+lnx1=0Ch. 4.6 - Solve the given equation for x. ln(x+1)ln(x2)=1Ch. 4.6 - Solve the given equation for x....Ch. 4.6 - Differentiate. y=ln[(x+5)(2x1)(4x)]Ch. 4.6 - Differentiate. y=ln[(x+1)(2x+1)(3x+1)]Ch. 4.6 - Differentiate. y=ln[(1+x)2(2+x)3(3+x)4]Ch. 4.6 - Differentiate. y=ln[e2x(x3+1)(x4+5x)]Ch. 4.6 - Differentiate. y=ln[xex2+1]Ch. 4.6 - Prob. 38ECh. 4.6 - Differentiate. y=ln(x+1)4ex1Ch. 4.6 - Differentiate. y=ln(x+1)4(x3+2)x1Ch. 4.6 - Prob. 41ECh. 4.6 - Prob. 42ECh. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Prob. 47ECh. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Prob. 51ECh. 4.6 - Prob. 52ECh. 4.6 - Prob. 53ECh. 4.6 - Prob. 54ECh. 4 - State as many laws of exponents as you can recall.Ch. 4 - Prob. 2FCCECh. 4 - Prob. 3FCCECh. 4 - Prob. 4FCCECh. 4 - Prob. 5FCCECh. 4 - Prob. 6FCCECh. 4 - Prob. 7FCCECh. 4 - Prob. 8FCCECh. 4 - Prob. 9FCCECh. 4 - Prob. 10FCCECh. 4 - Prob. 11FCCECh. 4 - Prob. 12FCCECh. 4 - Prob. 13FCCECh. 4 - Prob. 14FCCECh. 4 - Calculate the following. 274/3Ch. 4 - Calculate the following. 41.5Ch. 4 - Prob. 3RECh. 4 - Prob. 4RECh. 4 - Calculate the following. (25/7)14/5Ch. 4 - Prob. 6RECh. 4 - Prob. 7RECh. 4 - Calculate the following. 40.240.3Ch. 4 - Simplify the following. (ex2)3Ch. 4 - Simplify the following. e5xe2xCh. 4 - Simplify the following. e3xexCh. 4 - Simplify the following. 2x3xCh. 4 - Simplify the following. (e8x+7e2x)e3xCh. 4 - Simplify the following. e5x/2e3xexCh. 4 - Solve the following equations for x. e3x=e12Ch. 4 - Solve the following equations for x. ex2x=e2Ch. 4 - Solve the following equations for x. (exe2)3=e9Ch. 4 - Solve the following equations for x. e5xe4=eCh. 4 - Differntiate the following functions. y=10e7xCh. 4 - Differntiate the following functions. y=exCh. 4 - Differentiate the following functions. y=xex2Ch. 4 - Differentiate the following functions. y=ex+1x1Ch. 4 - Differntiate the following functions. y=eexCh. 4 - Differntiate the following functions. y=(x+1)e2xCh. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=xeCh. 4 - The graph of the functions f(x)=ex24x2 is shown in...Ch. 4 - Show that the function in Fig. 1 has a relative...Ch. 4 - Solve the following equations for t....Ch. 4 - Solve the following equations for t. et8e0.02t=0Ch. 4 - Solve the equation 42x=ex. [Hint: Express 2x as an...Ch. 4 - Solve the equation 3x=2ex. [Hint: Express 3x as an...Ch. 4 - Find the points on the graph of y=ex where the...Ch. 4 - Find the points on the graph y=ex+e2x where the...Ch. 4 - Determine the intervals where the function...Ch. 4 - Determine the intervals where the function...Ch. 4 - Find the equation of the tangent line to the graph...Ch. 4 - Show that the tangent lines to the graph of...Ch. 4 - Simplify the following expressions. e(ln5)/2Ch. 4 - Simplify the following expressions. eln(x2)Ch. 4 - Simplify the following expressions. lnx2lnx3Ch. 4 - Simplify the following expressions. e2ln2Ch. 4 - Simplify the following expressions. e5ln1Ch. 4 - Simplify the following expressions. [elnx]2Ch. 4 - Solve the following equations for t. tlnt=eCh. 4 - Solve the following equations for t. ln(ln3t)=0Ch. 4 - Solve the following equations for t. 3e2t=15Ch. 4 - Solve the following equations for t. 3et/212=0Ch. 4 - Solve the following equations for t. 2lnt=5Ch. 4 - Solve the following equations for t. 2e0.3t=1Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=xlnxCh. 4 - Differentiate the following functions. y=ln(5x7)Ch. 4 - Differentiate the following functions. y=ln(9x)Ch. 4 - Differentiate the following functions. y=(lnx)2Ch. 4 - Differentiate the following functions. y=(xlnx)3Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=xlnxxCh. 4 - Differentiate the following functions. y=e2ln(x+1)Ch. 4 - Differentiate the following functions. y=ln(lnx)Ch. 4 - Differentiate the following functions. y=1lnxCh. 4 - Differentiate the following functions. y=exlnxCh. 4 - Differentiate the following functions. y=ln(x2+ex)Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=ln|2x+1|Ch. 4 - Differentiate the following functions. y=ln(ex2x)Ch. 4 - Differentiate the following functions. y=lnx3+3x23Ch. 4 - Differentiate the following functions. y=ln(2x)Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=ln|x1|Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=ln(1ex)Ch. 4 - Differentiate the following functions....Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Prob. 80RECh. 4 - Prob. 81RECh. 4 - Prob. 82RECh. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Prob. 84RECh. 4 - Prob. 85RECh. 4 - Prob. 86RECh. 4 - Prob. 87RECh. 4 - Health Expenditures The health expenditures (in...
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- Total marks 15 4. : Let f R2 R be defined by f(x1, x2) = 2x²- 8x1x2+4x+2. Find all local minima of f on R². [10 Marks] (ii) Give an example of a function f R2 R which is neither bounded below nor bounded above, and has no critical point. Justify briefly your answer. [5 Marks]arrow_forward4. Let F RNR be a mapping. (i) x ЄRN ? (ii) : What does it mean to say that F is differentiable at a point [1 Mark] In Theorem 5.4 in the Lecture Notes we proved that if F is differentiable at a point x E RN then F is continuous at x. Proof. Let (n) CRN be a sequence such that xn → x ЄERN as n → ∞. We want to show that F(xn) F(x), which means F is continuous at x. Denote hnxn - x, so that ||hn|| 0. Thus we find ||F(xn) − F(x)|| = ||F(x + hn) − F(x)|| * ||DF (x)hn + R(hn) || (**) ||DF(x)hn||+||R(hn)||| → 0, because the linear mapping DF(x) is continuous and for all large nЄ N, (***) ||R(hn) || ||R(hn) || ≤ → 0. ||hn|| (a) Explain in details why ||hn|| → 0. [3 Marks] (b) Explain the steps labelled (*), (**), (***). [6 Marks]arrow_forward4. In Theorem 5.4 in the Lecture Notes we proved that if F: RN → Rm is differentiable at x = RN then F is continuous at x. Proof. Let (xn) CRN be a sequence such that x → x Є RN as n → ∞. We want F(x), which means F is continuous at x. to show that F(xn) Denote hn xnx, so that ||hn||| 0. Thus we find ||F (xn) − F(x) || (*) ||F(x + hn) − F(x)|| = ||DF(x)hn + R(hn)|| (**) ||DF(x)hn|| + ||R(hn) || → 0, because the linear mapping DF(x) is continuous and for all large n = N, |||R(hn) || ≤ (***) ||R(hn)|| ||hn|| → 0. Explain the steps labelled (*), (**), (***) [6 Marks] (ii) Give an example of a function F: RR such that F is contin- Total marks 10 uous at x=0 but F is not differentiable at at x = 0. [4 Marks]arrow_forward
- 3. Let f R2 R be a function. (i) Explain in your own words the relationship between the existence of all partial derivatives of f and differentiability of f at a point x = R². (ii) Consider R2 → R defined by : [5 Marks] f(x1, x2) = |2x1x2|1/2 Show that af af -(0,0) = 0 and -(0, 0) = 0, Jx1 მx2 but f is not differentiable at (0,0). [10 Marks]arrow_forward(1) Write the following quadratic equation in terms of the vertex coordinates.arrow_forwardThe final answer is 8/π(sinx) + 8/3π(sin 3x)+ 8/5π(sin5x)....arrow_forward
- Keity x२ 1. (i) Identify which of the following subsets of R2 are open and which are not. (a) A = (2,4) x (1, 2), (b) B = (2,4) x {1,2}, (c) C = (2,4) x R. Provide a sketch and a brief explanation to each of your answers. [6 Marks] (ii) Give an example of a bounded set in R2 which is not open. [2 Marks] (iii) Give an example of an open set in R2 which is not bounded. [2 Marksarrow_forward2. (i) Which of the following statements are true? Construct coun- terexamples for those that are false. (a) sequence. Every bounded sequence (x(n)) nEN C RN has a convergent sub- (b) (c) (d) Every sequence (x(n)) nEN C RN has a convergent subsequence. Every convergent sequence (x(n)) nEN C RN is bounded. Every bounded sequence (x(n)) EN CRN converges. nЄN (e) If a sequence (xn)nEN C RN has a convergent subsequence, then (xn)nEN is convergent. [10 Marks] (ii) Give an example of a sequence (x(n))nEN CR2 which is located on the parabola x2 = x², contains infinitely many different points and converges to the limit x = (2,4). [5 Marks]arrow_forward2. (i) What does it mean to say that a sequence (x(n)) nEN CR2 converges to the limit x E R²? [1 Mark] (ii) Prove that if a set ECR2 is closed then every convergent sequence (x(n))nen in E has its limit in E, that is (x(n)) CE and x() x x = E. [5 Marks] (iii) which is located on the parabola x2 = = x x4, contains a subsequence that Give an example of an unbounded sequence (r(n)) nEN CR2 (2, 16) and such that x(i) converges to the limit x = (2, 16) and such that x(i) # x() for any i j. [4 Marksarrow_forward
- 1. (i) which are not. Identify which of the following subsets of R2 are open and (a) A = (1, 3) x (1,2) (b) B = (1,3) x {1,2} (c) C = AUB (ii) Provide a sketch and a brief explanation to each of your answers. [6 Marks] Give an example of a bounded set in R2 which is not open. (iii) [2 Marks] Give an example of an open set in R2 which is not bounded. [2 Marks]arrow_forward2. if limit. Recall that a sequence (x(n)) CR2 converges to the limit x = R² lim ||x(n)x|| = 0. 818 - (i) Prove that a convergent sequence (x(n)) has at most one [4 Marks] (ii) Give an example of a bounded sequence (x(n)) CR2 that has no limit and has accumulation points (1, 0) and (0, 1) [3 Marks] (iii) Give an example of a sequence (x(n))neN CR2 which is located on the hyperbola x2 1/x1, contains infinitely many different Total marks 10 points and converges to the limit x = (2, 1/2). [3 Marks]arrow_forward3. (i) Consider a mapping F: RN Rm. Explain in your own words the relationship between the existence of all partial derivatives of F and dif- ferentiability of F at a point x = RN. (ii) [3 Marks] Calculate the gradient of the following function f: R2 → R, f(x) = ||x||3, Total marks 10 where ||x|| = √√√x² + x/2. [7 Marks]arrow_forward
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