Answers
Given: A surveyor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the tree as 75.8 degrees.
Required: To determine how accurately the angle must be measured if the percent error in estimating the tree's height is less than 5%.
Explanation: To estimate the angle, we will use the trigonometric ratio
[tex]tanx=\frac{h}{50}\text{ ...\lparen1\rparen}[/tex]where h is the tree's height, and x is the angle of elevation to the top of the tree.
Hence we get
[tex]\begin{gathered} h=50\cdot(tan75.8\degree) \\ h=197.59\text{ feet} \end{gathered}[/tex]Now differentiating equation 1, we get
[tex]sec^2xdx=\frac{1}{50}dh[/tex]We can write the above equation as:
[tex]sec^2x\cdot\frac{xdx}{x}=\frac{h}{50}\cdot\frac{dh}{h}\text{ ...\lparen2\rparen}[/tex]Also, it is given that the error in estimating the tree's height is less than 5%.
So
[tex]\frac{dh}{h}=0.05[/tex]Also, we need to convert the angle x in radians:
[tex]x=1.32296\text{ rad}[/tex]Putting these values in equation (2) gives:
[tex]\frac{dx}{x}=\frac{197.59}{50}\cdot\frac{cos^2(1.32296)}{1.32296}\cdot0.05[/tex]Solving the above equation gives:
[tex]\begin{gathered} \frac{dx}{x}=3.9518\cdot0.04548551012\cdot0.05 \\ =0.008987\text{ radians} \end{gathered}[/tex]Let
[tex]d\theta\text{ be the error in estimating the angle.}[/tex]Then,
[tex]\lvert{d\theta}\rvert\leq0.008987\text{ radians}[/tex]Final Answer:
[tex]\lvert{d\theta}\rvert\leq0.008987\text{ radians}[/tex]Related Questions
Answer parts a through E for the function shown below
Answers
Solution
We are given the function
[tex]f(x)=x^3+4x^2-x-4[/tex]First, Let us do the simplification or factorization
[tex]\begin{gathered} f(x)=x^2(x+4)-1(x+4) \\ f(x)=(x^2-1)(x+4) \\ f(x)=(x-1)(x+1)(x+4) \end{gathered}[/tex](a).
The coefficient of x^3 is positive
(b).
So basically, we set f(x) = 0 to get the x - intercepts
[tex]\begin{gathered} f(x)=(x-1)(x+1)(x+4) \\ (x-1)(x+1)(x+4)=0 \\ x=1,-1,-4 \end{gathered}[/tex]The x - intercepts are
[tex]x=1, -1, -4[/tex]The graph of f(x) is also given below
If the 10 letters are {aa,aa,aa,aa,bb,bb,cc,cc RR,RR} are available and all 10 of them are to be selected without replacement,what is the number of different permutations?
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In order to calculate the number of permutations, first we start with the factorial of the number of letters.
There are 10 letters, so we start with the factorial of 10.
Then, we need to check the number of repetitions. Each repetition will be a factorial in the denominator:
[tex]x=\frac{10!}{a!\cdot b!\operatorname{\cdot}...}[/tex]We have four repetitions of aa, two repetitions of bb, two repetitions of cc and two repetitions of RR, therefore the final expression for the number of permutations is:
[tex]x=\frac{10!}{4!2!2!2!}[/tex]Calculating this expression, we have:
[tex]x=\frac{10\operatorname{\cdot}9\operatorname{\cdot}8\operatorname{\cdot}7\operatorname{\cdot}6\operatorname{\cdot}5\operatorname{\cdot}4!}{4!\operatorname{\cdot}2\operatorname{\cdot}2\operatorname{\cdot}2}=\frac{10\operatorname{\cdot}9\operatorname{\cdot}8\operatorname{\cdot}7\operatorname{\cdot}6\operatorname{\cdot}5}{8}=18900[/tex]Therefore there are 18900 permutations.
If you borrow $100 for 3 years at anannual interest rate of 9%, howmuch will you pay altogether?
Answers
We are to determine the amount that you have pay back after borrowing a principal amount ( P ) for ( t ) number of years which is compounded annualy at rate ( R ).
You borrowed a principal amount of:
[tex]P\text{ = \$100}[/tex]The time duration for which we have borrowed the money for is:
[tex]t\text{ = 3 years}[/tex]The annual interest rate coumpounded each year is:
[tex]R\text{ = 9\% / year}[/tex]Step 1: Determine the simple interest that accumulated at the end of ( t ) years.
The folllowing formula is used to determine the simple interest that the borrower has to pay once the period of borrowing/lending is over i.e ( t ) years.
The simple interest is the proportional rate of interest ( R ) and the initial borrowed/loaned amount called principal amount ( P ).
[tex]\text{Simple Interest ( I ) = }\frac{P\cdot R\cdot t}{100}[/tex]Use the above simple interest formula ( I ) by plugging in the respective values as follows:
[tex]\text{Simple Interest ( I ) = }\frac{100\cdot9\cdot3}{100}\text{ = \$27}[/tex]Therefore, the total amount of interest that the borrower must pay as an extra ( over the borrowed amount ) is $27.
Step 2: Determine the total amount that is to be returned/paid to the lender
The total amoun that is to be paid by the borrower ( you ) to the lender is the principal amount borrowed ( P ) and the amount of interest accumulated for the contractual time period i.e ( I ).
[tex]\begin{gathered} \text{Total amount to be paid = P + I} \\ \text{Total amount to be paid = \$100 + \$27} \\ \text{Total amount to be paid = }127 \end{gathered}[/tex]Therefore, the amount that you need to pay altogether is:
[tex]\textcolor{#FF7968}{127}\text{\textcolor{#FF7968}{ dollars}}[/tex]Can you please help me to answer the question #46
Answers
Part A
S(0) = 1116 - 4.04(0) (Replacing h=0)
S(0)= 1116 (Multiplying)
The answer is 1116 ft/s
Part B
S(10) = 1116 - 4.04(10) (Replacing h=10)
S(10) = 1116 - 40.4 (Multiplying)
S(10)= 1075.06
The answer is 1075.06 ft/s
Part C
S(30) = 1116 - 4.04(30) (Replacing h=30)
S(30) = 1116 - 121.1 (Multiplying)
S(30)= 994.9 (Subtracting)
The answer is 994.9 ft/s.
There are 130 people in a sport centre.
76 people use the gym
60 people use the swimming pool.
32 people use the track.
23 people use the gym and the pool.
8 people use the pool and the track.
20 people use the gym and the track.
6 people use all three facilities.
Given that a randomly selected person
uses the gym and the track, what is
the probability they do not use the
swimming pool?
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The probability is 0.57
What is meant by probability?
Probability is a discipline of mathematics that deals with appropriate units of how probable an event is to occur or how likely a statement is to be true. The probability of an occurrence is a number ranging from zero and 1, where 0 denotes the event's feasibility and 1 represents certainty. The greater the likelihood of an occurrence, the more probable it will occur. Tossing a fair (unbiased) coin is a basic example. Because the coin is fair, the two possibilities ("heads" and "tails") are equally likely; the chance of "heads" equals the probability of "tails," and because no other outcomes are possible, the probability of either "heads" or "tails" is 1/2.
Probability of using the pool = 97/225 = 0.43
Probability that they do not use the swimming pool = 1 - 0.43 = 0.57
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An object moves in simple harmonic motion with period 6 seconds and amplitude 4cm. At time =t0 seconds, its displacement d from rest is 0cm, and initially it moves in a negative direction. Give the equation modeling the displacement d as a function of time t.
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The general function for describing the displacement from the mean position in harmonic motion is:
[tex]d(t)=A\cdot\sin (\frac{2\pi}{T}\cdot t+\phi)\text{.}[/tex]Where:
• A is the amplitude,
,• T is the period,
,• φ is initial phase displacement.
From the statement, we know that:
• the amplitude is 4 cm,
,• at time t = 0 its displacement d from the rest is 0 → d(t = 0) = 0,
,• initially, it moves in a negative direction.
s
A pottery factory purchases a continuous belt conveyor kiln for $68,000. A 6.3% APR loan with monthly payments is taken out to purchase the kiln. If the monthly payments are $765.22, over what term is this loan being paid?
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Based om the cost of the continuous belt conveyor kiln and the monthly payments, as well as the APR of the loan, the term this loan will be paid is 120 months or 10 years.
How to find the term of the loan?When given the cost of a loan, the APR, and the monthly payments, you can find out the term of the loan by using the NPER function on a spreadsheet.
The Rate would be:
= 6.3% / 12 months in a year
= 0.525%
The Pmt is the payment of $765.22. This amount should be in negatives.
The Present Value or Pv should be the loan amount of $68,000.
The term on the loan would then be 120 months which is 10 years.
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Graph the line y = 5x - 1, then name the slope and y-intercept by looking at the graph. What is m= and what is b= and how do I graph this what are the points ?
Answers
Answer:
Step-by-step explanation:
Slope-intercept form: y = mx + b
The 'm' in this formula means slope. The 'b' means the y-intercept.
y = 5x - 1
m = 5.
b = -1.
Now that we have identified the slope and the y-intercept, we can graph the equation.
When graphing these kinds of equations, always start at the y-intercept.
The y-intercept is -1, so we start from there and move up 5 and right 1 repeatedly.
Remember, slope = rise/run. We rise 5, and we run 1.
5 can also be represented as a fraction: [tex]\frac{5}{1}[/tex]
Let me know if you have any questions.
can you please give me any examples on how to do this
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we can take two numbers of the sequence and subtract them to see the difference
so
[tex]1.9-1.2=0.7[/tex]the sequence adds 0.7 each step
the next 3 terms are
[tex]3.3+0.7=4[/tex][tex]4+0.7=4.7[/tex][tex]4.7+0.7=5.4[/tex]A can of diced tomatoes has a height of 11.5 cm and a diameter of 10 cm. What is the volume of the can? Use 3.14 for pie.DO NOT round your answer.
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Answer:
902.75 cubic cm.
Explanation:
Given a can with:
• Height, h = 11.5 cm
,• Diameter = 10 cm
A can is in the shape of a cylinder; and the volume of a cylinder is calculated using the formula:
[tex]V=\pi r^2h[/tex]First, find the radius by dividing the diameter by 2.
[tex]r=\frac{10}{2}=5\;cm[/tex]Next, substitute r=5, h=11.5 and π=3.14 into the formula given above:
[tex]\begin{gathered} V=3.14\times5^2\times11.5 \\ =902.75\text{ cubic cm} \end{gathered}[/tex]The volume of the can is 902.75 cubic cm.
The graph shows the proportional relationship between the number of gems collected and the number of levels that have been completed in a video game.
Graph with x axis labeled game levels and y axis labeled gems collected. A line begins at 0 comma 0 and goes through points 6 comma 420 and 8 comma 560.
Determine the constant of proportionality for the relationship.
p = 70
p = 140
p equals 2 over 140
p = 0.0143
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Answer: P = 70
Step-by-step explanation:
p = 70 because on the graph everytime the y number is the x number multiplied by 70.
70 x 2 = 140
70 x 4 = 280
70 x 6 = 420
70 x 8 = 560
70 x 10 = 700.
Heres the chart for proof
The constant of proportionality (p) for this proportional relationship is equal to: A. p = 70.
How to determine the constant of proportionality?In Mathematics, the graph of any proportional relationship is characterized by a straight line because as the values on the x-axis increases or decreases, the values on the y-axis increases or decreases simultaneously.
Mathematically, a proportional relationship can be represented by the following equation:
y = px
Where:
p is the constant of proportionality.y represents the gems collected.x represents the game levels.Next, we would determine the constant of proportionality (p) for the data points on this graph as follows:
p = y/x
p = 420/6 = 560/8
p = 70.
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7^2 × 7^8. 7^a------------ = -------- = 7^b7^4 7^4
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We have to find the values of a and b:
[tex]\frac{7^2\cdot7^8}{7^4}=\frac{7^a}{7^4}=7^b[/tex]We can use the laws of exponents to write:
[tex]\begin{gathered} 7^2\cdot7^8=7^a \\ 7^{2+8}=7^a \\ 7^{10}=7^a \\ 10=a \end{gathered}[/tex]Then, we can solve for b as:
[tex]\begin{gathered} \frac{7^a}{7^4}=7^b \\ 7^{a-4}=7^b \\ a-4=b \\ 10-4=b \\ 6=b \end{gathered}[/tex]Answer: a=10 and b=6
which point lies on the wall with point slope equation y+5=2(x+8)
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The slope intercept form of equation is given as
(y - y1) = m(x - x1)
Where m = slope
From the equation: y + 5 = 2(x + 8)
Equate y + 5 = 0 and x + 8 = 0
y + 5 = 0
y = 0- 5
y = -5
For x + 8 = 0
x + 8 = 0
x = 0 - 8
x = -8
Hence, the point is (-8, -5)
The answer is (-8, -5)
which of the following lines is perpendicular to the equation given below?
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Given data:
The given equation of the line is y=-2x+8.
The slope of the given line is -2.
The slope of the line perpendicular to it is,
[tex]\begin{gathered} m\times-2=-1 \\ m=\frac{1}{2} \end{gathered}[/tex]The standard equation of the line is,
[tex]y=mx+c[/tex]Here, m is the slope of the line.
The second option can be written as,
[tex]\begin{gathered} x-2y=8 \\ 2y=x-8 \\ y=\frac{1}{2}x-4 \end{gathered}[/tex]Thus, option (B) is correct.
why you can always solve a right triangle if you know the measures of one side and one acute angle.
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In a right triangle, one angle is always 90.
If you know one acute angle, you automatically know the other (3rd) angle.
3 angles are solved.
Now, comes the sides.
If you already know 1 side, you can easily know another side by using the basic trig identities SIN, COS, or TAN.
When you know 2 sides, the 3rd side can always be find using:
• pythagorean theorem, or
,• again, trigonometric ratios (sin, cos, tan).
You play a game where you toss a die. If the die lands on a 6, you win $6. It costs $2 toplay. Construct a probability distribution for your earnings. Find your expected earnings.
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SOLUTION
Now from the question, if the die lands on 6, I win $6. So probability of landing on 6 is
[tex]\frac{1}{6}\text{ since a die has 6 faces }[/tex]Since I will pay $2 to play, we subtract this from $6 that we will win.
And probability of losing becomes
[tex]\frac{5}{6}\text{ }[/tex]The table becomes
From the table the expected earnings is calculated as
[tex]\begin{gathered} E=\sum_^xP(x) \\ =4(\frac{1}{6})-2(\frac{5}{6}) \\ =\frac{4}{6}-\frac{10}{6} \\ =-\frac{6}{6} \\ =-1 \end{gathered}[/tex]Hence expected earnings is -$1
If Omar still needs 458How much does he need after saving for 5 weeks
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(a) Setting A(w)=458, we get:
[tex]800-18w=458.[/tex]Subtracting 800 from the above equation we get:
[tex]\begin{gathered} 800-18w-800=458-800, \\ -18w=-342. \end{gathered}[/tex]Dividing the above equation by -18 we get:
[tex]\begin{gathered} \frac{-18w}{-18}=\frac{-342}{-18}, \\ w=19. \end{gathered}[/tex]Therefore Omar has been saving for 19 weeks.
(b) Recall that to evaluate a function at a given value, we substitute the variable by the given value.
Evaluating A(w) at w=5 we get:
[tex]A(5)=800-18\cdot5.[/tex]Simplifying the above result we get:
[tex]\begin{gathered} A(5)=800-90 \\ =710. \end{gathered}[/tex]Answer:
(a) 19.
(b) $710.
The graph below and to the left shows the time of sunsets occurring every other day during September in a certain town. The graph at the lower right shows the time of sunsets on either the 21st or 22nd day of each month for an entire year in the same town. The vertical axis is scaled to reflect hours after midnight. Round to 4 decimal places. a) Find a linear model for the data in the graph at the left. Include units to your variables. b) Find a cosine model for the data in the graph to the right. Include units to your variables,
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A) Given the points (1,18.35) and (29,17.5), we can find the linear model with the following formulas:
[tex]\begin{gathered} \text{slope:} \\ m=\frac{y_2-y_1}{x_2-x_1}=\frac{71.5-18.35}{29-1}=\frac{-0.85}{28}=-0.03 \\ \text{equation of the line:} \\ y-y_1=m(x-x_1) \\ \Rightarrow y-18.35=-0.03(x-1)=-0.03x+0.03 \\ \Rightarrow y=-0.03x+0.03+18.35=-0.03x+18.38 \\ y=-0.03x+18.38 \end{gathered}[/tex]therefore, the linear model is y = -0.03x+18.38
B)We have the general cosine model:
[tex]y(t)=A+B\cos (\omega(t-\phi))[/tex]Where A is the vertical shift, B is the amplitude, w is the frequency and phi is the phase shift.
First, we can find the vertical shift with the following formula:
[tex]A=\frac{y_{\max }+y_{\min }}{2}[/tex]in this case, we have that the maximum value for y is 19.47 and the minimum value for y is16.18, then:
[tex]A=\frac{19.47+16.18}{2}=17.825[/tex]next, we can find the amplitud with the following formula:
[tex]B=y_{\max }-A[/tex]We have then:
[tex]B=19.47-17.825=1.645[/tex]Now, notice that the graph will repeat every 356 values for t, then, for the frequency we have the following expression:
[tex]\omega=\frac{2\pi}{356}=\frac{\pi}{178}[/tex]To find the phase shift, notice that for the point (172,19.47), we have the following:
[tex]\begin{gathered} y(172)=19.47 \\ \Rightarrow17.825+1.645\cos (\frac{\pi}{178}(172-\phi))=19.47 \\ \Rightarrow1.645\cos (\frac{\pi}{178}(172-\phi))=1.645 \\ \Rightarrow\cos (\frac{\pi}{178}(172-\phi))=1 \end{gathered}[/tex]notice that if the cosine equals 1, then its argument must equal to 0, then, we have:
[tex]\begin{gathered} \frac{\pi}{178}(172-\phi)=0 \\ \Rightarrow172-\phi=0 \\ \Rightarrow\phi=172 \end{gathered}[/tex]we have that the phase shift is phi = 172, then, the final cosine model is:
[tex]y(x)=17.825+1.465\cos (\frac{\pi}{178}(x-172))[/tex]Why was math created
Answers
SOLUTION:
Step 1:
In this quesdtion, we are meant to explain the topic:
Why was Math created?"
Step 2:
The details of the solution are as follows:
1. Mathematics is a body of knowledge and knowledge and practice, that is derived from the contributions of thinkers throughout the ages and across the globe.
2. It gives us a way to understand patterns, to quantify relationships, and to predict the future.
3. Math helps us understand the world — and we use the world to understand math.
4. Excellent for your brain: Creative and analytical skills are highly desired by employers.
5. It has a lot of real-world applications.
6. It helps in better problem-solving skills.
7. Mathematics is needed in almost every career and profession.
8. Mathematics helps understand the world better.
9. Mathematics is a universal language.
10. Numbers help us understand the world, and Mathematics helps us understand numbers.
The real-life applications of Mathematics are endless.
We are surrounded by numbers, equations and algorithms – especially in this age of data science, with huge data sets that can only be understood through statistical models and analysis.
Answer:
Maths was invented to understand the world and to make measurements, do calculations etc. make and measure shapes, measure angles, and to use these things in real life. The Egyptians used the Pythagoras theorem to accurately make their pyramids.
Construct a probability distribution for a discrete random variable uses the probability experiment of tossing a coin three times. Consider the random variable for the number of heads
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Answer:
Explanation:
By building a tree diagram we can find the theoretical probability of each number of heads when tossing three coins.
A manufacturing process has a 70% yield, meaning that 70% of the products
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Answer:
are acceptable and 30% are defective.
Step-by-step explanation:
Write the slope-intercept form of the equation of the line graphed on the coordinate plane.
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The slope-intercept form is:
[tex]y\text{ = mx + b}[/tex]We have to find these coefficients. To do that we have to choose two points in the graph and apply the following formula. I will use (0,1) and (-1,-1). The formula is:
[tex]y-yo\text{ = m(x-xo)}[/tex]The formula of the coefficient 'm' is:
[tex]m\text{ = }\frac{y2-y1}{x2-x1}[/tex]Let's substitute the points into the formula above to find the value of m. Then we use one of the points to find the slope-intercept form of the equation:
[tex]m\text{ = }\frac{-1-1}{-1-0}=2[/tex]Applying it to the second equation using the point (0,1):
[tex]y-1=2(x-0)[/tex][tex]y=2x+1[/tex]Answer: The slope-intercept form of the equation will be 2x+1.
Sports Authority marks up New Balance sneakers $30 and sells them for $109. Markup is on cost. What are the cost and percent markup?
Answers
Answer: $79 and percentage is 36%
Find the coordinates of the other endpoint of the segment, given its midpoint and one endpoint. (Hint: Let (x,y) be the unknown endpoint. Apply the midpoint formula,and solve the two equations for x and y.)midpoint (1,17), endpoint (-5,13)
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The coordinates of a midpoint of a line delimited by two endpoints is:
[tex]\begin{gathered} x_m=\frac{x_1+x_2}{2} \\ y_m=\frac{y_1+y_2}{2} \end{gathered}[/tex]Where (xm,ym) are the coordinates of the midpoint, (x1,y1) are the coordinates of the first endpoint and (x2,y2) are the coordinates of the second endpoint. We want to find (x2,y2), therefore:
[tex]\begin{gathered} 1=\frac{-5+x_2}{2} \\ 2=-5+x_2 \\ x_2=2+5=7 \end{gathered}[/tex][tex]\begin{gathered} 17=\frac{13+y_2}{2} \\ 34=13+y_2 \\ y_2=34-13 \\ y_2=21 \end{gathered}[/tex]The coordinates of the endpoint two are (7,21).
Enter the explicit and recursive equations for the sequence 2, -4, -10, -16 Please HELP
Answers
The explicit and recursive forms of the arithmetic sequence are f(n) = 2 - 6 · (n - 1) and f(n) = f(n - 1) - 6, f(1) = 2, respectively.
How to derive equations for the elements of an arithmetic sequence
In this problem we need to find the explicit and recursive equations for an arithmetic sequence, whose definitions are described below:
Explicit form
f(n) = a + r · (n - 1)
Recursive form
f(n) = f(n - 1) + r, f(1) = a
Where:
a - First element of the sequence.r - Common difference.n - Index of the n-th element of the sequence.If we know that a = 2, r = - 6, then the explicit and recursive forms of the sequence are:
Explicit form
f(n) = 2 - 6 · (n - 1)
Recursive form
f(n) = f(n - 1) - 6, f(1) = 2
The first four elements of the sequence generated by the formulas are 2, - 4, - 10, - 16.
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A baseball stadium has 50,100 seats. Each ticket for a seat costs $30. Tara created a function to model this situation and drew the graph of the function, where y represents profit from ticket sales, in dollars, given the number of tickets sold, x.
Is the graph function correct? why or why not?
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The graph as shown in the image is the correct graph of the function.
What is the correct graph of the function?A function shows a mathematical relationship. We would need to look at the graph very closely so as to know weather or not the graph as it has been shown is the correct graph that is befitting of the function must be a straight line graph.
Clearly, the slope of the graph would be positive and beginning from the origin because the number of tickets that is sold is increasing just and the amount of the tickets is increasing. Thus the graph follows the general equation of a straight line; y = mx + c
All these goes to show that what we have befits the function.
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1 + c + 1.4 = c + 2.4I need help
Answers
1 + c + 1.4 = c + 2.4
c + 2.4 = c + 2.4
c = c + 2.4 - 24
c = c
What is the solution to the equation below? 6x= x + 20 O A. x = 4 B. X = 20 C. x = 5 D. No Solutions
Answers
Simplify the equation 6x = x +20 to obtain the value of x.
[tex]\begin{gathered} 6x=x+20 \\ 6x-x=20 \\ 5x=20 \\ x=\frac{20}{5} \\ =4 \end{gathered}[/tex]So answer is x = 4
Option A is correct.
Solve the quadratic equation using any algebraic method. Show all work that leads to your answer.
6x² + 23x + 20 = 0
Answers
Answer:
x = - [tex]\frac{5}{2}[/tex] , x = - [tex]\frac{4}{3}[/tex]
Step-by-step explanation:
6x² + 23x + 20 = 0 ( factorise the left side )
consider the factors of the product of the coefficient of the x² term and the constant term which sum to give the coefficient of the x- term.
product = 6 × 20 = 120 and sum = 23
the factors are + 8 and + 15
use these factors to split the x- term
6x² + 8x + 15x + 20 = 0 ( factor the first/second and third/fourth terms )
2x(3x + 4) + 5(3x + 4) = 0 ← factor out (3x + 4) from each term
(3x + 4)(2x + 5) = 0
equate each factor to zero and solve for x
2x + 5 = 0 ⇒ 2x = - 5 ⇒ x = - [tex]\frac{5}{2}[/tex]
3x + 4 = 0 ⇒ 3x = - 4 ⇒ x = - [tex]\frac{4}{3}[/tex]
Find all critical points of the function f(x) = x^3 + 5x^2 - 7x - 3.The critical point(s) is(are) =
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We are given:
[tex]f(x)=x^3+5x^2-7x-3[/tex]Now, we know that in order to determine the critical points we derivate and the derivative is then equal to 0, that is:
[tex]f^{\prime}(x)=3x^2-10x-7=0[/tex]Now, we solve for x, that is:
[tex]3x^2+10x-7=0\Rightarrow x=\frac{-(10)\pm\sqrt[]{(10)^2-4(3)(-7)}}{2(3)}[/tex][tex]\Rightarrow\begin{cases}x=-\frac{5+\sqrt[]{46}}{3}\Rightarrow x\approx-3.9 \\ \\ x=\frac{-5+\sqrt[]{46}}{3}\Rightarrow x\approx0.6\end{cases}[/tex]So, the critical points of the function are:
[tex]\begin{cases}x=-\frac{5+\sqrt[]{46}}{3} \\ \\ x=\frac{-5+\sqrt[]{46}}{3}\end{cases}[/tex]Now, we determine the y-components of the points, that is:
[tex]\begin{cases}f(-\frac{5+\sqrt[]{46}}{3})=(-\frac{5+\sqrt[]{46}}{3})^3+5(-\frac{5+\sqrt[]{46}}{3})^2-7(-\frac{5+\sqrt[]{46}}{3})-3\Rightarrow f(-\frac{5+\sqrt[]{46}}{3})=41.03608735 \\ \\ f(\frac{-5+\sqrt[]{46}}{3})=(\frac{-5+\sqrt[]{46}}{3})^3+5(\frac{-5+\sqrt[]{46}}{3})^2-7(\frac{-5+\sqrt[]{46}}{3})-3\Rightarrow f(\frac{-5+\sqrt[]{46}}{3})=-5.184235498\end{cases}[/tex]So, the two critical points are:
[tex](-\frac{5+\sqrt[]{46}}{3},41.03608735)[/tex]and:
[tex](\frac{-5+\sqrt[]{46}}{3},-5.184235498)[/tex]This can be seing as follows:
