calculus ! find the equation of the normal line to the curve `f(x) = (1)/(x)` at the point (1,1)
Answers
The slope of the tangent line at the point x = a of the function f(x) is f'(a).
We are given the function:
[tex]f(x)=\frac{1}{x}[/tex]Computing the first derivative:
[tex]f^{\prime}(x)=-\frac{1}{x^2}[/tex]The slope of the tangent line at (1, 1), that is, where x = 1 is:
[tex]f^{\prime}(1)=-\frac{1}{1^2}=-1[/tex]The tangent line and the normal line are perpendicular to each other. If their respective slopes are m1 and m2, then:
[tex]m_1\cdot m_2=-1[/tex]We have calculated m1 = -1, calculate m2:
[tex]m_2=-\frac{1}{m_1}=-\frac{1}{-1}=1[/tex]Now we know the slope of the normal line. We need to find its equation. Use the point-slope formula:
y - k = m(x - h)
Where m is the known slope and (h, k) is a point of the line. We are given the point (1, 1), thus:
y - 1 = 1(x - 1) = x - 1
Adding 1:
y = x
Answer: D. x
Related Questions
Which of the following expressions is equivalent to the one shown below?(-5.4)How do I solve this and what’s the answer ?
Answers
Option B
Using the law of indices
[tex]undefined[/tex]help meeeeeeeeeeeeeeeeeeeeeeeeeee
Answers
Answer: 9.7 seconds
Step-by-step explanation:
[tex]16t^2=1503\\\\t^2 =\frac{1503}{16}\\\\t=\sqrt{1503/16} \text{ } (t > 0)\\\\t \approx 9.7[/tex]
if dyllan makes 7 out of 10 free throws on any given day what is the probability he will make his next free throw? ( write your answers as a precent.)
Answers
Answer:
70%
Explanation:
The probability can be calculated as the number of free throws that Dylan makes divided by the total number of free throws.
So, if Dylan makes 7 out of 10 free throws, the probability can be calculated as:
[tex]P=\frac{7}{10}=0.7[/tex]Then, to know the probability as a percent, we need to multiply 0.7 by 100% to get:
P = 0.7 x 100% = 70%
Therefore, the answer is 70%
Find the values of each variable.
Answers
Step-by-step explanation:
x + y = 180° (linear pair)
Also, x = 3y (corresponding angles are equal)
So, 4y = 180
y = 45°
x = 135°
what is the volume of the cone? use 3.14 for Pi and round to the nearest hundredth
Answers
The volume of the cone:
[tex]V\text{ = }50.24in^3[/tex]Explanations:The radius of the cone, r = 2 in
The height of the cone, h = 4 in
The volume, V, of a cone is given by the formula:
[tex]V\text{ = }\pi r^2h[/tex]Substituting r = 2, h = 4, and π = 3.14 into the formula for calculating the volume shown above:
[tex]\begin{gathered} V=3.14\text{ }\times2^2\text{ }\times\text{ 4} \\ V\text{ = 3.14 }\times\text{ 4 }\times\text{ 4} \\ V\text{ = }50.24in^3 \end{gathered}[/tex]What is -2(-x+5y-4) (simplify equation)
Answers
Using the distributed property, the simplified form of -2(-x+5y-4) is 2x-10y+8.
In the equation question,
The given expression is -2(-x+5y-4).
We have to simplify the given expression.
We simplify the given expression using the distributed method.
In the distributed method we have to multiply the number outside the bracket to all the terms that are inside the bracket.
As a(b+c) = ab+bc
In the given question there is -2 outside the bracket and -x+5y-4 is inside the bracket.
To simplify the given expression we multiply -2 to each term inside the bracket.
-2(-x+5y-4) = (-2)×(-x)+(-2)×5y-(-2)×4
Simplifying
-2(-x+5y-4) = 2x+(-10)y-(-8)
Simplifying the bracket
-2(-x+5y-4) = 2x-10y+8
Hence, the simplified form of -2(-x+5y-4) is 2x-10y+8.
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Divide $70 into these ratios 1:2:4
Answers
Answer:
Step-by-step explanation:
1+2+4=7
70/7=10
10 x 1=10 10 x 2=20 10 x 4=40
10:20:40
hi. This is an equation on some math homework and im not sure how to do it. Could you help?Write an equation of the line that is parallel to y=16 and passes through point (−7,43)
Answers
From the given equation y=16, it implies that the line has no slope.
Hence, the slope is zero (0).
From the given point:
[tex]\begin{gathered} (-7,43) \\ x_1=-7,y_1=43 \end{gathered}[/tex]The equation of the line is given by the equation:
[tex]m=\frac{y-y_1}{x-x_1}[/tex][tex]\begin{gathered} 0=\frac{y-43}{x-(-7)} \\ 0=\frac{y-43}{x+7} \\ \text{cross}-\text{multiply} \\ y-43=0 \\ y=43 \end{gathered}[/tex]Hence, the equation of the line is y=43
2. An account is opened with $28000 earning12% interest compounded yearly. What is thebalance in the account after 4 years?
Answers
The principal is $28000 with rate 12% compounded yearly .
To determine the balance after 4 years,
[tex]SI=\frac{P\times R\times T}{100}[/tex][tex]SI=\frac{28000\times12\times4}{100}[/tex][tex]SI=13440[/tex]The balance in the account after 4 years is the sum of principal and simple interest.
[tex]A=\text{ 28000}+13440[/tex][tex]A=41440[/tex]Thus, the balance in the account after 4 years is 41440 dollar.
What is the solution of the system of equations?
13x-6y=2
3x-4y=-10
(Write as ordered pair)
Answers
Step-by-step explanation:
let's multiply the first equation by 2 and the second equation by -3, and then we add them together :
26x - 12y = 4
-9x +12y = 30
-------------------------
17x 0 = 34
x = 2
and now we use one of the original equations to solve for y :
3×2 - 4y = -10
6 - 4y = -10
-4y = -16
y = 4
the solution is (2, 4).
NASA is painting the nose cone of a sounding rocket with a special sealant which reduces the air-drag on the rocket. If they need to do two coats of the sealant, how many square feet are they painting? Use π = 3.14.A. 56.5 ft²B. 27.3 ft²C. 35.3 ft²D. 28.3 ft²
Answers
In order to determine the number of required square feet to paint the nose cone, use the following formula for the surface area of a cone without the base:
[tex]S=\pi rl[/tex]where
r: radius of the base of the cone = 3 ft
l: slant height of the cone = 6 ft
replace the previous values of the parameters into the formula for S and simplify:
[tex]S=(3.14)(3ft)(6ft)=56.5ft^2[/tex]Hence, 56,5ft^2 are required
Help me this is very hard
Answers
By applying the distributive property of multiplication, an expansion of this mathematical expression can be written as 3(x - 2) = 3x - 6.
What is the distributive property of multiplication?In Mathematics, the distributive property of multiplication states that when the sum of two (2) or more addends are multiplied by a particular numerical value, the same result and output would be obtained as when each addend is multiplied respectively by the same numerical value, and the products are added together.
Mathematically, the distributive property of multiplication can be represented by this mathematical expression:
a(b + c) = ab + ac.
a(b - c) = ab - ac
By applying the distributive property of multiplication to the given mathematical expression, we have the following:
3(x - 2) = 3(x) - [3(2)]
3(x - 2) = 3x - 6
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Triangle STU has side lengths s = 34.0 and u = 40.5. Angle T has a measure of 49°. What
is the length of side t?
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Using cosine rule, the length of side t is 31.5 unit
Cosine RuleCosine rule states that the square of the length of any side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them.
The formula is given as
t² = s² + u² -3su cos(T)
let's define our variables
t = ?s = 34u = 40.5T = 49 degreessubstituting the values into the formula
t² = 34² + 40.5² -2(34)(40.5)cos49
t² = 1156 + 1640.25 - 1806.79
t² = 989.46
t = √989.46
t = 31.5
The length of t is 31.5 unit
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I really need help I'll take a picture of the question
Answers
To plot the graph of the function:
[tex]h(x)=8|x+1|-1[/tex]According to your graphing software, we need to find the smallest value of the function, and any point at the positive slope side of the function. Let's start with the minimum point. Let's use the general formula for an absolute value function:
[tex]f(x)=a|x+b|+c[/tex]The coefficients 'b' and 'c' determinates how much the function was translated from the origin. The 'b' determinates in the x-axis, and the 'c' determinates in the y-axis.
If the 'b' is positive, the function translated to the left, if it is negative the function translated to the right.
If the 'c' is positive the function went up, and if it is negative it went down.
Now let's analyze our h(x) function. Our coefficients are:
[tex]\begin{cases}b=1 \\ c=-1\end{cases}[/tex]Which means our minimum point is
[tex](-1,-1)[/tex]Now, we just need to calculate any point to the right of x = -1.
Let's calculate for x = 0.
[tex]h(0)=8|0+1|-1=8-1=7[/tex]The second point is (0,7)
Mr. Kilgore made a pot of gumbo with 72 ounces of okra and 2 lbs of shrimp. He made another pot
of gumbo with the same amount of okra, but three times as many lbs of shrimp. How many ounces
of okra did he use per lb of shrimp in the second pot of gumbo
Answers
Answer:
12
Step-by-step explanation:
72 / 2 =36
36 / 3 = 12
12, 24, 36, 48, 60, 72.
Use the graph, which shows the profit,y, in thousands of dollars, of a company in a given year, t, where t represents the number of years since 1980.linear graph with negative slope with points (15,150) and (25,130)Find the linear function y, where y depends on t, the number of years since 1980. Type your function in slope intercept form (y=mx+b) without any commas or spaces and be sure to use a lowercase t for your variable.The linear function is y=Answer
Answers
First, locate any two point from the line.
(15, 150) and (25, 130)
x₁ = 15 y₁= 150
x₂ = 25 y₂ = 130
[tex]\text{Slope(m)}=\frac{130-150}{25-15}[/tex][tex]=\frac{-20}{10}[/tex][tex]=-2[/tex]Next, is to find the intercept.
Substitute x=15 y=150 and m=-2 into y=mx + b and solve for b
That is;
150 = 15(-2) + b
150 = -30 + b
Add 30 to both-side of he equation
150 + 30 = b
180 = b
Form the equation by substituting m=-2 and b=180 into y=mx + b
y = -2x + 180
Hence, the line function is y = -2x + 180
find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval.
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Remember that
If f is continuous over [a,b] and differentiable over (a,b), then there exists c∈(a,b) such that
[tex]f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}[/tex]In this problem, we have the function
[tex]f(x)=\frac{9}{x^3}[/tex]over the interval [1,3]
so
f(a)=f(1)=9/(1)^3=9
f(b)=f(3)=9/(3)^3=1/3
substitute
[tex]f^{\prime}(c)=\frac{\frac{1}{3}-9}{3-1}=\frac{-\frac{26}{3}}{2}=-\frac{26}{6}=-\frac{13}{3}[/tex]Find out the first derivative f'(x)
[tex]f^{\prime}(x)=-\frac{27}{x^4}[/tex]Equate the first derivative to -13/3
[tex]\begin{gathered} -\frac{27}{x^4}=-\frac{13}{3} \\ \\ x^4=\frac{27*3}{13} \\ \\ x=1.58 \end{gathered}[/tex]therefore
The value of c is 1.58 (rounded to two decimal places)find the measure of NE and MD PLEASE 5th time i need help
Answers
The measure is 5.
From the question, we have
E is the circumcenter. The distance from circumcenter to all of their vertices from that triangle are the same, so
NE =EP
⇒13x -28 = 6x +7
⇒13x -6x = 28 +7
⇒7x = 35
⇒x= 5
Subtraction:
Subtraction represents the operation of removing objects from a collection. The minus sign signifies subtraction −. For example, there are nine oranges arranged as a stack (as shown in the above figure), out of which four oranges are transferred to a basket, then there will be 9 – 4 oranges left in the stack, i.e. five oranges. Therefore, the difference between 9 and 4 is 5, i.e., 9 − 4 = 5. Subtraction is not only applied to natural numbers but also can be incorporated for different types of numbers.
The letter "-" stands for subtraction. Minuend, subtrahend, and difference are the three numerical components that make up the subtraction operation. A minuend is the first number in a subtraction process and is the number from which we subtract another integer in a subtraction phrase.
Complete question: Find value of x, measure of NE,MD?
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Mora is using a map to find the distance between her house and her cousin's house. She knows that 2 centimeters on the map represent 300 miles. On the map, the distance between her house and her cousin's house is 20 centimeters.
What is the distance, in miles, between Mora's house and her cousin's house?
Answers
The distance, in miles, between Mora's house and her cousin's house will be 3000 miles as "2 centimeters on the map represent 300 miles. On the map, the distance between her house and her cousin's house is 20 centimeters."
What is unitary method?The unitary method involves calculating the value of a single unit, from which we can calculate the values of the necessary number of units. A single or distinct unit is referred to by the word unitary. Therefore, the goal of this method is to establish values in relation to a single unit. The unitary method, for instance, can be used to calculate how many kilometers a car will travel on one litre of gas if it travels 44 km on two litres of fuel. The unitary method is a way of finding any required quantity's value by first finding the value of the unit (one) quantity.
here,
2 cm=300 miles
1 cm=150 miles
so,
20 cm=
150*20=3000 miles
Mora's residence and her cousin's residence are separated by 3000 miles in miles "On the map, a centimeter equals 300 miles. Her house is 20 centimeters from her cousin's house, according to the map."
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A population numbers 11,000 organisms initially and decreases by 9% each year.
Answers
Answer:
Step-by-step explanation:
11000*0.91=10010
10010*0.91=9109.1(approximately 9109)
keep going until you find the year you want
Write 250 metres as a fraction of a kilometre.
Answers
250 m / 1000 = 0.25km
The fraction would be ²⁵⁰⁄₁₀₀₀ or ¼ simplified.
Hope this helps!
a random sample of 2 measurements is taken from the following population of values: 1, 2, 4, 5, 8. what is the probability that the range of the sample is 7?
Answers
The probability of the sample required where the range is 7 is 0.1 .
The range of a sample is the difference of the maximum and minimum values.
The range of the sample to be 7, the maximum and minimum values should be 8 and 1.
The outcomes required are (8,1).
The total probability of the sample for 2 measured values:
P=[tex]C_{5,2}[/tex]
P=5!/(2!(5-2)!)
P=5!/(2![tex]\times[/tex]3!)
P=20/2
P=10
The probability of the outcome desired,
=(required outcomes)/(total outcomes)
=1/10
=0.1
The probability of the outcome required is 0.1 .
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If both pairs of opposite sides of a quadrilateral are congruent, then thequadrilateral is a parallelogram.A. TrueB. False
Answers
Answer:
True
Explanation:
PLEASE HELP
(x,y)→(__,__)
Answers
Answer: since the distance between f and g is 20 units and is then reduced to 4 units long, it's scale factor would be 0.2
Step-by-step explanation:
Sorry if you get this wrong, i tried really hard on this and watched a video
The ratio of the number of apple trees to the number of lemon trees in a farm is 9:5. There are 45 lemon trees in the farm. How many apple trees are there in the farm?
Answers
5x9=45
9x9=81
you’re welcome
A watermelon is placed on a digital scale. The scale reads 10.5. What could units be?
Answers
The units of weight of watermelon on the digital scale could be kilogram, gram and pounds.
According to the question,
We have the following information:
A watermelon is placed on a digital scale. The scale reads 10.5.
We know that there are several units of measurements for every physical quantity. For example, the units for measuring length are kilometers, meters, centimeters, feet, inches and many more.
In the same way, there are many units for measurements of weight. However, in most of the cases for the weight on a digital scale, it can be kilogram, gram or pound.
So, it could 10.5 kg or 10.5 g or 10.5 pounds.
Hence, the units of the weight of watermelon on the digital scale could be kilogram, gram and pounds.
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Learning Diagnostic Analytics Math Skill plans Recommendations Fifth grade > Z.11 Parts of a circle Q7N The radius of a circle is 12 feet. What is the circle's diameter? feet Submit
Answers
Given:
The radius of the circle = 12 feet
The circle diameter = two times of the radius
so, the diameter = 2 * 12 = 24 feet
So, the answer is 24 feet
At any time t (in hours), there are 216(t +18) of Type A bacteria in a sample and 362t + 8 of Type B bacteria in a sample. After how many hours will the number of type A bacteria be less than the type B bacteria?
Answers
1. The given situation can be described in the following inequality:
[tex]216^{(t+18)}<36^{(2t+8)}[/tex]2. To solve the prevous inequality, proceedas follow:
write 216 as 6^3 and 36 as 6^2, then, apply log_6 both sides:
[tex]\begin{gathered} 6^{3(t+18)}<6^{2(2t+8)} \\ \log _66^{3(t+18)}<\log _66^{2(2t+8)} \\ 3(t+18)<2(2t+8) \end{gathered}[/tex]then, solve for t, as follow:
[tex]\begin{gathered} 3t+54<4t+16 \\ 3t-4t<16-54 \\ -t<-38 \\ t>38 \end{gathered}[/tex]Hence, from t = 38 hours the number of type A bacteria will be less than the type B bacteria
3. In a number line, you obtain:
What transformations map the figure onto itself? Select all that apply.
A.
Reflection over line y
B.
Reflection over point P
С.
Rotation 180 degrees
D.
Reflection over line m
E.
Rotation 270 degrees
Answers
Answer:
Step-by-step explanation:
C because its right
2. A rock club's profit from booking local bands depends on the ticket price. Using past receipts, the owners find that
the profit p can be modeled by the function p = -15t² + 600t + 50, where t represents the ticket price in dollars.
a. What price yields the maximum profit?
b. What is the maximum profit?
Answers
Based on the given function,
a) The price that yields the maximum profit is $20.
b) The maximum profit is $6,050
Function
Function defines the relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
Given,
A rock club's profit from booking local bands depends on the ticket price. Using past receipts, the owners find that the profit p can be modeled by the function p = -15t² + 600t + 50, where t represents the ticket price in dollars.
Here we need to find the price that yields maximum profit and the amount of maximum profit.
In order to find the price that yield the maximum profit, we have to find the value of t, that can be obtained by using the formula,
t = -b/2a
According to the given function the values of a = -15 and the value of b = 600
Apply these values on the given function then we get,
t = -600/ 2(-15)
t = -600/-30
t = 20
Therefore, the price that yields the maximum profit is $20.
Now, we need to find the maximum profit, by apply the value of t as 20 in the given function then we get,
p = -15(20)² + 600(20) + 50
P = -15(400) + 12000 + 50
p = -6000 + 12000 + 50
p = 6000 + 50
Therefore, the maximum profit is $6,050.
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