Linearization is the idea in calculus where if you zoom in really close onto a section of the graph, the curve and the tangent line will be extremely similar to each other.
We can use this idea and approximate nearby values to the tangent line.
The parent graph of a square root function is [tex]f(x)=\sqrt{x}[/tex].
There are two methods in which we can calculate the approximate value of [tex]\sqrt{101}[/tex] by using linearization techniques.
Method 1:Write equation of tangent line.Find the value of y at x = 101.Let's use the nearest perfect square to [tex]\sqrt{101}[/tex], which is [tex]\sqrt{100}[/tex]. When drawing the graph of the square root function, we use the point (100, 10) because where x = 100, y = 10 on a square root function.
Now, we have a point that the tangent line passes through, and we want to find the slope.
We can do so by taking the derivative of [tex]f(x) =\sqrt{x}[/tex].
[tex]f'(x)=x^\frac{1}{2}[/tex] [tex]f'(x)=\frac{1}{2} x^-^\frac{1}{2}[/tex] [tex]f'(x)=\frac{1}{2x^\frac{1}{2} }[/tex] [tex]}f'(x)=\frac{1}{2\sqrt{x} }[/tex]Plug in x = 100 to find the slope of the tangent line here.
[tex]f'(100)=\frac{1}{2\sqrt{100} }[/tex] [tex]f'(100)=\frac{1}{2(10)}=\frac{1}{20}[/tex]The slope of the tangent line at x = 100 is 1/20. Now, we can use the point and the slope of the tangent line to determine the equation for the tangent line.
(100, 10), m = 1/20
Point-slope equation:
[tex]y-y_1=m(x-x_1)[/tex] [tex]y-(10)=\frac{1}{20}(x-(100))[/tex] [tex]y=\frac{1}{20}x-5+10[/tex] [tex]y=\frac{1}{20}x+5[/tex]To find the value of y at x = 101, plug this into the tangent line equation.
[tex]y=\frac{1}{20}(101)+5[/tex] [tex]y=\frac{101}{20}+5[/tex] [tex]y=\frac{101}{20} +\frac{100}{20} = \frac{201}{20}[/tex]201/20 is equal to 10.05. Let's estimate the error of this calculation to the actual value of [tex]\sqrt{101}[/tex], inputted into a calculator.
We get [tex]\sqrt{101}=10.04987562[/tex], which is extremely close to 10.05. Our estimate using linearization only has an error of 0.00012.
Method 2:You may have noticed that the first method uses little calculus, apart from taking the derivative of f(x).
The more "calculus-y" method involves finding the change in y in terms of the change of x.
We know that dy/dx is the the change in y due to a change in x, so we can use this to find the change in y when x = 101.
[tex]\frac{dy}{dx} =\frac{1}{2\sqrt{x} }[/tex]Solve for dy by mutliplying dx to both sides of the equation.
[tex]dy=\frac{1}{2\sqrt{x} } \cdot dx[/tex]The change in x is 1, because 101 - 100 = 1.
We want to find the change in y when x = 100, and has a change of 1. Therefore, we have:
[tex]dx=1[/tex] [tex]x=100[/tex]Plug these values into the equation [tex]dy=\frac{1}{2\sqrt{x} } \cdot dx[/tex] and solve for dy.
[tex]dy=\frac{1}{2\sqrt{200} } \cdot 1[/tex] [tex]dy=\frac{1}{2(10)}[/tex] [tex]dy=\frac{1}{20}[/tex]The change in y when x = 100 with a change of 1 is 1/20. We know when x = 100 the y-value is 10. Therefore, when x = 101, the change in y is 1/20, so y = [tex]10\frac{1}{20} = \frac{201}{20} = 10.05[/tex].
You can see that we get the same answer with either method.
X^2-5x+6=0 quadratic form
Answer:
x = 3,2
Step-by-step explanation:
solve for x by finding a,b,c and then apply the quadratic formula.
Answer:x
=
−
3
x
=
−
2
Explanation:
The quadratic formula is:
⇒
x
=
−
b
±
√
b
2
−
4
a
c
2
a
for a quadratic of the form
a
x
2
+
b
x
+
c
.
We have
a
=
1
,
b
=
5
, and
c
=
6
.
x
=
−
5
±
√
5
2
−
4
(
1
)
(
6
)
2
(
1
)
x
=
−
5
±
√
25
−
24
2
x
=
−
5
±
√
1
2
x
=
−
5
±
1
2
Hence,
x
=
−
6
2
=
−
3
and
x
=
−
4
2
=
−
2
I'm not sure why you wanted to use the quadratic formula, but you could just factor the quadratic:
x
2
+
5
x
+
6
=
(
x
+
3
)
(
x
+
2
)
=
0
x
+
3
=
0
→
x
=
−
3
and
x
+
2
=
0
→
x
=
−
2
Step-by-step explanation:
If f(x) = x - 4 and g(x) = 3x + 5, find (f + g)(x).
a) 4x + 9
b) 2x + 9
c) 2x - 1
d) 4x + 1
Answer:
4x+9
Step-by-step explanation:
x-4 +3x+5
x+3x+5+4=4x+9
WILL GIVE BRAINLIEST
Consider the line that passes through each pair of given points, and decide what type of slope the line has.
(2, 4) and (5, 1)
(-7, 8) and (-7,0)
(6,-3) and (-4,-3)
(3,5) and (-1,2)
Answer:
2.) undefined slope 3.) zero slope (i'm not sure what 1 and 4 are, sorry!)
Step-by-step explanation:
undefined slope has the same x-coordinate and the denominator becomes 0. it also is a vertical line. for a zero slope, it has the same y-coordinate and the numerator is 0. this is a horizontal line. *this is my first response so if i am wrong, that is embarrassing*
What is the image point of (1, 4) after a translation left 5 units and down 5
units?
Answer:
(-4,-1)
All you have to do to find it is subtract x by 5 and subtract y by 5. :)
if three fourth of a number is one-thenths, what is the number please show with steps first person answer u will get branliest!!!
Let x be the number we are searching for.
(3/4)x = 1/10
How many factors of 100 are also multiples of 5?
Let us write the factors of 100. A factor of 100 can only be a multiple of five if it is divisible by five.
Divisibility rule of 5: A number can only be divisible by five if the ones digit of the number is a five or 0.Factors of 100 ⇒ numbers divisible by 100 ⇒ 1, 2, 4, 5, 10, 20, 25, 50, 100
When checking each factor of 100;
1 is not a multiple of 5 ⇔ not divisible by 5.2 is not a multiple of 5 ⇔ not divisible by 5.4 is not a multiple of 5 ⇔ not divisible by 5.5 is a multiple of 5 ⇔ divisible by 5.10 is a multiple of 5 ⇔ divisible by 5.20 is a multiple of 5 ⇔ divisible by 5.50 is a multiple of 5 ⇔ divisible by 5.100 is a multiple of 5 ⇔ divisible by 5.⇒ Five factors are multiples of 5.
Therefore, five factors of 100 are multiples of 5.
What is the equation of the line shown in the graph?
Drag and drop the expressions to write the equation of the line in slope-intercept form.
y = Response area + Response area
A function graph of a line with two points (-3,-2) and (-1,2) with an x axis of negative five to five and a y axis of negative five to five
9514 1404 393
Answer:
y = 2x + 4
Step-by-step explanation:
You have the graph, so you can read the slope and y-intercept from it. The line goes up 2 units for each unit to the right, so the slope is ...
slope = rise/run = 2/1 = 2
The y-intercept is where the graph crosses the y-axis, at y=4.
The slope-intercept form of the equation for a line is ...
y = mx + b
where m is the slope (2) and b is the y-intercept (4). Your equation is ...
y = 2x +4
Answer: First blank; -x
second; -2
Step-by-step explanation:
which line is parallel to y=8x+11?
Answer:
Any line with a slope of 8 is parallel to y = 8x + 11. Some examples are:
y = 8x + 5
y = 8x + 1000
y = 8x +9
It doesn't matter what the y-intercept as long as the slope remains the same for a parallel line.
Answer:
Like the other person said, any line that has the same slope (8) will be parallel.
Step-by-step explanation:
Which of the following is a prime number between 40 and 50? A. 41 B. 45 C. 44 D. 49
Answer:
A.41
Step-by-step explanation:
carla, bev, and ashley took their unit 2 math test. carla made an 80 and bev made a 75. what was ashleys score on the test if their average was 78.
Answer:
Ashley = 79
Step-by-step explanation:
[tex]Average=\frac{Carla+Bev+Ashley}{3}=78\\\\\frac{80+75+Ashley}{3}= 78 \\\\\frac{155+Ashley}{3}=78\\\\\frac{155+Ashley}{3}*\frac{3}{1} =78*3\\\\155+Ashley=234\\[/tex]
Subtract 155 from both sides;
Ashley = 79
On a coordinate plane, 2 straight lines are shown. The first solid line has a negative slope and goes through (negative 2, 3) and (0, negative 1). Everything to the left of the line is shaded. The second dashed line has a negative slope and goes through (0, 2) and (1, 0). Everything to the right of the line is shaded.
Which inequality pairs with y≤−2x−1 to complete the system of linear inequalities represented by the graph?
y −2x+2
y 2x−2
Answer:B i think......
Step-by-step explanation:
Answer: It's y>-2x+2
I don’t know how to set it up nor do it please help me
Answer:
The value of x is 3
Step-by-step explanation:
∵ Quadrilateral ABCD is congruent to quadrilateral JKLM
∴ AB = JK and BC = KL
∴ CD = LM and AD = JM
∵ BC = 8x + 7
∵ KL = 31
∵ BC = KL
→ Equate their right sides
∴ 8x + 7 = 31
→ Subtract 7 from both sides
∵ 8x + 7 - 7 = 31 - 7
∴ 8x = 24
→ Divide both sides by 8 to find x
∴ [tex]\frac{8x}{8}[/tex] = [tex]\frac{24}{8}[/tex]
∴ x = 3
∴ The value of x is 3
This table shows the times it took four students to complete a 40-meter race.
Race Times
Student
Time
(in seconds)
Joe
13.07
Justin
13.6
Maria
13.06
Melinda
13.7
Which statement about these times is true?
A: 13.7 > 13.07
B: 13.07 > 13.6
C: 13.6 > 13.7
D: 13.06 > 13.07
what is one thousandth less than 0.029
Answer:
0.0009
Step-by-step explanation:
Sam's dad says he will pay a fraction of his mobile phone bill. Would Sam rather his dad paid 1/2 of the bill or the 1/3?
Answer:
matters
Step-by-step explanation:
if sam doesnt want his dad to spend a lot of money on him then 1/3
but if sam wants to pay less then 1/2
1/2 is more then 1/3
Whats the answer help please
Need now thank you
Correct answer
Will give brainliest
Answer:The slope is 6/7
Step-by-step explanation:
when finding the slope you will need to move the 6x to the other side of the equation by subtracting which gives you 7y=14-6x and all that is left is to divide by 7 which gives y=14-6/7y
Answer:
6x - 7y = 14
⇔ 6x - 14 = 7y
⇔ 6/7.x - 2 = y
=> the slope is 6/7
Step-by-step explanation:
A number t multiplied by −4 is at least −25.
Answer:
[tex]t\leq 6.25[/tex] in decimal form
[tex]t\leq \frac{25}{4}[/tex] in fractional form
Step-by-step explanation:
-4 times a number t is greater than or equal to - 25. This is what they mean by at least.
[tex]-4t \geq -25[/tex]
Solve for t.
[tex]t\leq \frac{-25}{-4}[/tex] , we flip the inequality because we are dividing by a negative number.
[tex]t\leq 6.25[/tex]
Find the slope of the line that passes through (5, 13) and (9, 10).
Simplify your answer and write it as a proper fraction, improper fraction, or integer.
Answer:
The slope between (5, 13) and (9, 10) is:
[tex]m=-\frac{3}{4}[/tex]Step-by-step explanation:
Given the points
(5, 13)(9, 10)Finding the slope between (5, 13) and (9, 10)
[tex]\mathrm{Slope}=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m=\frac{10-13}{9-5}[/tex]
[tex]m=-\frac{3}{4}[/tex]
Therefore, the slope between (5, 13) and (9, 10) is:
[tex]m=-\frac{3}{4}[/tex]Please solve with elimination
5x + 9y =-5
3x -8y = -3
help I will give brainliest
please show work
Answer:
Step-by-step explanation:
5x+9y=-3 Multiply both sides by 8
40x+72y=-24
-53x-8y=-3 Multiply both sides by 9
-477x -72y= -27
sum the equations vertically to eliminate one variable
40x+72y= -24
-477x-72y= -27 (the y cancels out )
-477x =-51 (-24 + -27=-51)
-477x=-51 / -477 divide both sides by -477
x =51/437
5x 51/437=9y= -3 substitute the value of x
y= -174 /437
(x,y) = (51/437, -174/437)
Which list is in order from least to greatest?
A. 2/3, 29%, 1/4 , 0.06
B. 2/3, 1/4, 29%, 0.06
C. 0.06, 1/4, 29%, 2/3
D. 1/4, 2/3, 0.06, 29%
Answer:
c
Step-by-step explanation:
2/3 converts to 0.66666667, 66.67%
1/4 converts to 0.25 , 25%
0.06 converts to 6%
So the order from least to greatest would be 0.06 , 1/4 , 29% , 2/3
Halla tres enteros impares consecutivos tales que 3 veces el entero del medio es 5 mas que la suma del primero y el tercero.
Responder:
3, 5, 7
Explicación paso a paso:
Sean enteros impares consecutivos:
x, x + 2, x + 4
3 (x + 2) = (x + x + 4) + 5
3x + 6 = (2x + 4 + 5)
3x + 6 = 2x + 9
3x - 2x = 9 - 6
x = 3
Por lo tanto,
x = 3; (x + 2) = 3 +2 = 5; x + 4 = .3 + 4 = 7
(3, 5 y 7)
Jeff ordered 5 pizzas for his daughters birthday
party. Each pizza has 8 slices. At the party, 50%
of the pizzas were eaten. When the party was
over, Jeff ate 20% of what was left. How many
did Jeff eat?
Answer:
4 slices.
Step-by-step explanation:
5 x 8 = 40. 40 slices of pizza divded by 2 since 50% = 2.
20 slices, 20% of 20 slices or 20 in general is 4
3 n - 5 p + 2 n = 10 p for n
Answer:
What do you mean
Step-by-step explanation:
Kellie ran 1/5 mile in 1 1/2 minutes at this rate , how long would it take her to run 1 mile
Answer:
PLZ MARK BRAINLIEST
The answer is 7.5 miles or 7 1/2.
Step-by-step explanation:
1 1/2 times 5= 7 1/2 or 7.5 miles
Answer:
7 1/2 min
Step-by-step explanation:
1.5 x 5 = 7.5 She ran 1 mile in 7 1/2 min
Find the area. Simplify your answer.
3x-3
3x+6
Answer:
3x^2 + 7x - 18
Step-by-step explanation:
if you know the answer to this PLEASE HELPPPP MEEE
Answer:
-2, -1, 1, 3
Step-by-step explanation:
Range is the amount so that is why it is the answer
PLEASE HELP! I WILL GIVE BRAINLIEST! pleeeaaaasssseee!
Solve by factoring
Answer:
n = -35/1301
Step-by-step explanation:
1225n+76n=1301n=-35
n = -35/1301
Alana runs 4 laps around the track in 10 minutes. If Alana runs 17 laps at the same rate how long does it take her
Answer:
42 minues 5 seconds to do 17 laps.
Step-by-step explanation:
it takes her 2 minuetes and 50 seconds to do 1 lap if you multiply 2.5x17=
42 munites and 5 secondesto do 17 laps
Answer:
It would take 42.5 min
Step-by-step explanation:
4/10 = 2.5(unit rate for every lap it takes 2.5 min)
17 x 2.5 = 42.5
Find the slope of the line through the given points.
(-9, -7) and (-11, -13)
use distributive property to rewrite 3×546
Answer:
1638
Step-by-step explanation:
3(500)+3(40)+3(6)=1500+120+18=1638