Answer:
[tex]\displaystyle \lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3} = \boxed{ \frac{1}{4} }[/tex]
General Formulas and Concepts:
Calculus
Limits
Limit Rule [Variable Direct Substitution]:
[tex]\displaystyle \lim_{x \to c} x = c[/tex]
Special Limit Rule [L’Hopital’s Rule]:
[tex]\displaystyle \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}[/tex]
Differentiation
DerivativesDerivative NotationDerivative Property [Addition/Subtraction]:
[tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Derivative Rule [Basic Power Rule]:
Derivative Rule [Chain Rule]:
[tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify given limit.
[tex]\displaystyle \lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3}[/tex]
Step 2: Find Limit
Let's start out by directly evaluating the limit:
[Limit] Apply Limit Rule [Variable Direct Substitution]:When we do evaluate the limit directly, we end up with an indeterminant form. We can now use L' Hopital's Rule to simply the limit:
[Limit] Apply Limit Rule [L' Hopital's Rule]:∴ we have evaluated the given limit.
___
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___
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Limits
Without using l'Hopitâl's rule, we can instead rationalize the numerator.
[tex]\dfrac{\sqrt{x+1} - 2}{x-3} = \dfrac{\left(\sqrt{x+1}-2\right)\left(\sqrt{x+1}+2\right)}{(x-3)\left(\sqrt{x+1}+2\right)} = \dfrac{(x+1)-4}{(x-3)\left(\sqrt{x+1}+2\right)} = \dfrac1{\sqrt{x+1}+2}[/tex]
We've removed the discontinuity at [tex]x=3[/tex], so the limand is continuous and
[tex]\displaystyle \lim_{x\to3} \frac{\sqrt{x+1}-2}{x-3} = \lim_{x\to3} \frac1{\sqrt{x+1}+2} = \frac1{\sqrt{3+1}+2} = \boxed{\frac14}[/tex]
helpppppppppppppppphhhh
Answer:
put a point on the 9 (the line on the left of 10) and on the -6 (the line to the left of -5).
a. 9 is located to the right of -6
b. use the symbol >
Step-by-step explanation:
please help me ...................
The figure is the combination of a square, a rectangle, and a triangle. Then the area of the figure will be 213 mm².
What is Geometry?It deals with the size of geometry, region, and density of the different forms both 2D and 3D.
The figure is the combination of a square, a rectangle, and a triangle. Then the area of the figure will be
Area of square = 4 x 4
Area of square = 16 mm²
Area of rectangle = 7 x 9
Area of rectangle = 53 mm²
Area of triangle = 0.5 x 24 x 12
Area of triangle = 144 mm²
Then the area if the figure will be
Area of the figure = 16 + 53 + 144
Area of the figure = 213 mm²
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g(1) = 0
=
lgn= ) +m
g(n) = g(n - 1) +n
=
-
n
g(2) =
Answer:
g(1) = 0
=
lgn= ) +m
g(n) = g(n - 1) +n
=
-
n
g(2) =12
Step-by-step explanation:
12345678910
I NEED HELP FAST GET BRAINLIEST AND 100 POINTS:
Galvin and Henry spend a certain amount of money from their money box each month to buy plants. The table (function 1) shows the relationship between the amount of money (y) remaining in Galvin’s money box and the number of months (x): Number of Months (x) Amount Remaining (in $) (y) 1 80 2 75 3 70 4 65 The equation (function 2) shows the relationship between the amount of money, y, remaining in Henry’s money box and the number of months, x: Function 2 y = −7x + 80 Which statement explains which function shows a greater rate of change? Function 2 shows a greater rate of change, because Henry spends $7 each month and Galvin spends $5 each month. Function 2 shows a greater rate of change, because Henry spends $80 each month and Galvin spends $15 each month. Function 1 shows a greater rate of change, because Galvin spends $15 each month and Henry spends $73 each month. Function 1 shows a greater rate of change, because Galvin spends $5 each month and Henry spends −$7 each month.
Answer:
Function 2 shows a greater rate of change, because Edwin spends $9 each month and Adam spends $7 each month.
I found this out because for Adam, you simply subtract the y-coordinates from each other and get 7 each time. Edwin clearly shows that he Spends $9 each month with the negative sign.
Step-by-step explanation:
Answer:
Function 2 shows a greater rate of change, because Henry spends $7 per month and Galvin spends $5 per month.
Step-by-step explanation:
Function 1
Money remaining in Galvin's money box:
[tex]\begin{tabular}{| c | c |}\cline{1-2} Number of Months (x) & Amount Remaining (in \$) (y)\\\cline{1-2} 1 & 80\\\cline{1-2} 2 & 75 \\\cline{1-2} 3 & 70 \\\cline{1-2} 4 & 65 \\\cline{1-2}\end{tabular}[/tex]
We can calculate the rate of change by using this formula:
[tex]\textsf{rate of change}=\dfrac{\textsf{change in y}}{\textsf{change in x}}[/tex]
[tex]\implies \textsf{rate of change}=\dfrac{75-80}{2-1}=\dfrac{-5}{1}=-5[/tex]
Therefore, The rate of change of function 1 is -5 which means that Galvin spends $5 per month.
Function 2
Money remaining in Henry's money box:
[tex]\sf y = -7x+80[/tex]
This is represented as a linear equation: y = mx + b
(where m is the slope or "rate of change" and b is the y-intercept of "initial value)
Therefore, the rate of change of function 2 is -7 which means that Henry spends $7 per month.
Conclusion
Function 2 shows a greater rate of change, because Henry spends $7 per month and Galvin spends $5 per month.
Robert is baking a cake. The recipe calls for 3 1/2 pounds of flour for every 1/4 cup of sugar. How many pounds of flour should Robert use for 2 cups of sugar??
uhhh what is this this looks very hard
Answer:
B. x ≤ 5
Step-by-step explanation:
This is an inequality problem. It can be solved by examining the arrow and the point it closes on. If the arrow is going left, then all values of x are going to be less than the point it closes on. If the arrow is going right, then all values of x are going to be greater than the point it closes on.
If the circle on the point the arrow is closed/shaded in, then that value will be included in the values of x, giving the inequality either a ≤ or ≥ sign depending on which way the arrow is pointing. If the circle on the point the arrow is open, then that value will not be included in the values of x. The sign will be either < or >.
---------------------------------------------------------------------------
For this problem, the arrow is going to the left of 5, which means all values of x are going to be less than 5. Because the circle is closed, that means that number 5 is included, or x can be equal to 5. If the circle was open, it wouldn't be included. So, the inequality is x ≤ 5.
hope this helps!
The beginning mean wage in a certain industry is $35,240.00. If the mean wage grows by 5.125%, what is the new mean wage?
Answer:
$37046.05
Step-by-step explanation:
new mean wage = beginning mean wage x (105.125%)
= 35240 x 105.125%
= 37046.05
Answer:
$37,046.05
Step-by-step explanation:
New mean wage :
Original mean wage (1 + % increase)$35,240.00 (1 + 5.125%)$35,240.00 (1.05125)$37,046.05Half dollar coins weigh 4 ounces and quarters weigh
0.2 pound, Laurel has 9 half dollar coins and 25 quarters in
her coin bank. Her coin bank weighs 2 pounds. Laurel lifts the
coin bank. How much does the bank and the coins weigh?
(A) 43 ounces
(C) 118 ounces
(B) 88 ounces
(D) 148 ounces
Answer:
A roll of quarters from the bank is worth $10 and weighs almost exactly half a pound, so 200 pounds works out to $4000.
Find the outer perimeter of this figure.
Round your answer to the nearest
hundredth. Use 3.14 to approximate pi.
Answer:
62.84 m
Step-by-step explanation:
6*3.14 = 18.84
12*2 = 24
10*2 = 20
total: 62.84
The solution is 62.84 m
The perimeter of the figure is P = 62.84 m
What is a Circle?
A circle is a closed two-dimensional figure in which the set of all the points in the plane is equidistant from a given point called “center”. Every line that passes through the circle forms the line of reflection symmetry. Also, the circle has rotational symmetry around the center for every angle
The perimeter of circle = 2πr
The area of the circle = πr²
where r is the radius of the circle
The standard form of a circle is
( x - h )² + ( y - k )² = r²,
where r is the radius of the circle and (h,k) is the center of the circle
Given data ,
Let the perimeter of the figure be represented as P
Now , the figure has a semi circle , a square and a triangle
The side of the square = 12m
So , the number of sides of square in the perimeter of the figure = 2 sides
So , perimeter of 2 sides = 2 x 12 = 24 m
Now , the 2 sides of the hypotenuse of the right triangle is also part of the figure , so
perimeter of 2 sides of the triangle = 2 x 10 = 20 m
Now , the total perimeter of figure = perimeter of the semi circle + 24 + 20
Perimeter of semi circle = 2πr / 2
The radius of the circle = 6 m
Substituting the value of r in the equation , we get
Perimeter of the semi circle = ( 2 x 3.14 x 6 ) / 2
Perimeter of the semi circle = 18.84 m
So , the perimeter of the figure P = 18.84 m + 24 m + 20 m
The perimeter of the figure P = 62.84 m
Therefore , the value of P is 62.84 m
Hence , The perimeter of the figure is P = 62.84 m
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Item 7 Lakima has a spinner divided into 5 equal sections that are labeled 1 though 5. She wants to compare the theoretical probability and the experimental probability of spinning an odd number. She spins the spinner 6 times and records the results in this list. {2,4,1,5,3,4} Drag and drop the answers into the boxes to correctly complete the sentences. The theoretical probability of spinning an odd number is equal to Response area. The experimental probability of spinning an odd number is equal to Response area. Therefore, the theoretical probability of spinning an odd number is Response area the experimental probability of spinning an odd number.
The theoretical probability of spinning an odd number is equal to 3/5. The experimental probability of spinning an odd number is equal to 1/2.
What is Probability?Probability helps us to know the chances of an event occurring.
[tex]\rm Probability=\dfrac{Desired\ Outcomes}{Total\ Number\ of\ outcomes\ possible}[/tex]
Given the spinner has 5 sections, each section is numbered between 1 to 5. Lakima also spins the spinner 6 times and records the results in this list. {2,4,1,5,3,4}. Therefore,
The theoretical probability of spinning an odd number is equal to 3/5.
The experimental probability of spinning an odd number is equal to 3/6=1/2.
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Answer: The theoretical probability of spinning an odd number is equal to 3/5. The experimental probability of spinning an odd number is equal to 1/2. Therefore, the theoretical probability of spinning an odd number is GREATER THAN the experimental probability of spinning an odd number.
Step-by-step explanation:
A shoe box has a volume of 64 cubic inches and a
surface area of 160 square inches. What is the surface
area of a similar shoe box if its volume is only 27 cubic inches? Show your work or explain how you got your answer
The surface area of a similar box given the ratio of volume to surface area of first box is 67.5 square inches
How to solve ratio?Box A:
volume = 64 cubic inchesSurface area = 160 square inchesBox B:
volume = 27 cubic inchesSurface area = xVolume : surface area
64 : 160 = 27 : x
64/160 = 27/x
64 × x = 160 × 27
64x = 4,320
x = 67.5 square inches
Therefore, the surface area of a similar box given the ratio of volume to surface area of first box is 67.5 square inches
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help please pleaseeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
Answer:
B, C, E
Step-by-step explanation:
the dependent variable depends on the independent variable
thus, p is dependent on t, p is the dependent variable, and t is the independent variable (B)
every month, Gloria paints 2 pieces
each month, t is increased by 1, and p is increased by 2 (C)
every month, Gloria paints 2 pieces
2 pieces every month = (2*t) pieces = total pieces = p (E)
Which parent function is represented by the graph
Explore rectangular and triangular prisms and their two-
Answer:24
Step-by-step explanation:
An irregularly shaped rock was lowered into a graduated cylinder holding a volume of water equal to 50 ml. The height of the water rose to 75 ml. If the mass of the stone was 250 g, what was its density?
5 grams per cubic centimeter
Which kind of triangle is shown? Help me
Answer: Right triangle
Step-by-step explanation: There is a 90 degree angle diagonal from the hypotenuse
Answer:
right triangle
Step-by-step explanation:
if you double the triangle, it becomes a rectangle
What is the area of ΔABC to the nearest tenth of a square inch?
PLEASE SHOW WORK!
Answer:
1440.1 in^2
Step-by-step explanation:
I drew a picture with work; hopefully this is the equation you're using in the lesson.
The circumference of a circle is 24π cm. What is the area, in square centimeters? Express your answer in terms of \piπ.
Given the value of the circumference of the circle, its area in terms of constant pi is 144πcm².
What is a circle?A circle is simply a closed 2-dimensional curved shape with no corners or edges.
The area and circumference of a circle is expressed mathematically as;
A = πr²C = 2πrWhere A is area, C is circumference, r is radius and π is constant pi.
Given the data in the question;
Circumference of the circle C = 24π cmArea A = ?First, we determine the radius of the circle.
24π cm = 2πr
r = 24π cm / 2π
r = 12cm
Now, we find the area.
A = πr²
A = π × (12cm)²
A = π × 144cm²
A = 144πcm²
Therefore, given the value of the circumference of the circle, its area in terms of constant pi is 144πcm².
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Using the data set below, what is the range?
4,6,6,7,7,8,8,8,9A
5
B
2
C
4
Answer:
A
Step-by-step explanation:
Subtract the minimum data value from the maximum data value to find the data range. In this case, the data range is
9 - 4 = 5
A roll of tape has a circumference of 19 centimeters. The circumference decreases 4 centimeters after you use some of the tape. What is the diameter of the roll after you use the tape?
QUICK PLS
Answer:
4.77 cm correct to the nearest hundredth.
Step-by-step explanation:
The new circumference = 19 - 4 = 15 cm.
Circumference C = π * diameter
So the diameter = C/π = 15/π = 4.7746 cm
For the region bounded by y=3x, the x-axis, and x=4, determine the volume when
The area bounded by the line y=3x from limits x=0 to x=4 will be 24.
What is integration?
Integration is defined as the adding up all the small units to find the whole unit.
It is given in the question that the line y=3x is making a region in a plane from the limits x= to x=4.
So by integrating the function we will get the area.
[tex]\int\limits^4_0 y=\int\limits^4_0 {3x} \, dx[/tex]
[tex]\int\limits^4_0 y=3 \int\limits^4_0 {x} \, dx[/tex]
[tex]\int\limits^4_0 y=[\dfrac{3x^2}{2}]^4_0[/tex]
[tex]\int\limits^4_0 y= \dfrac{(3\times (4)^2)}{2}[/tex]
[tex]\int\limits^4_0 y= 24[/tex]
hence the region bounded by the line y=3x will have the area of 24 square units.
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Jon has 18 apple and mia gives him 3 times more than he has how many dose jon have know and how many did mia give jon
Answer:
18x3=54
Mia gave Jon 54 apples
54+18=72
Now Jon has 72 apples
PLS HELP ASAP THANK YOU ;)
Answer:
x=2
Step-by-step explanation:
5x-5 = x+3
4x=8
x=2
Several students made comments about this image.
A triangle has angles B, C, D. The exterior angle to angle B is A.
Which comment is true?
“If I know the measures of A and B, I can subtract those measures from 180 to find the measures of the other two angles.”
“If I know the measure of C, I can find the measure of angle A because they are the same.”
“If I know the measures of angles C and B, I can find the measures of A and D.”
“If I know the measure of angle A, I can find the measure of angle D by subtracting from 180
Answer:
The third comment is the true one: "If I know the measures of angles C and B, I can find the measures of A and D."
This is because you can use the angle B to find angle A by subtracting it from 180, and you can find angle D by adding together C and B and subtracting that from 180.
I hope this helps!
Step-by-step explanation:
Answer with an algebraic equation!!!
Scott is 7 years older than his brother Mark. Their combined ages are 35. How old is each boy?
Answer:
Scott is 21 his brother is 14
Step-by-step explanation:
you know that scott is 7 years older than his unknown variable brother, so use what is given and form an alegbraic expression 2x+7=35
Jahiem is creating a decoration on the floor in the following pattern,
How many more tiles of each shape will he need so that the figure is symmetrical over the horizontal
and vertical lines?
There are 3 red Hexagones, 3 blue triangles and 5 orange rectangles needed to complete the symmetry of the given figure.
What is symmetry?Something is symmetrical when it is the same on both sides. A shape has symmetry if a central dividing line (a mirror line) can be drawn on it, to show that both sides of the shape are exactly the same.
The given figure will become symmetrical when 3 red Hexagones, 3 blue triangles and 5 orange rectangle added to the right portion.
Hence there are 3 red Hexagones, 3 blue triangles and 5 orange rectangles needed to complete the symmetry of the given figure.
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Some residuals for a scatterplot are given. for x = 3, the residual is –5. for x = 5, the residual is –1. for x = 7, the residual is 3. for x = 9, the residual is 0. for which value of x is the data point farthest from the line of best fit? 3 5 7 9
The value of x that is the data point farthest from the line of best fit is 3
How to determine the x value farthest from the line of best fit?The given parameters are:-
x Residual
3 -5
5 -1
7 3
9 0
Take the absolute value of the residuals
x Residual
3 5
5 1
7 3
9 0
The largest absolute value is 5
Hence, the value of x that is the data point farthest from the line of best fit is 3
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Answer:
answer is 3
Step-by-step explanation:
Find the measures of the missing angles. The triangle is not drawn to scale.
108
A=
B= 31
PLEASEE HELP
Answer:
41 degrees
Step-by-step explanation:
Angles in a triangle add to 180 degrees, so angle A is 180-108-31=41 degrees.
Estimate:
11 is 50% of what number?
A 20
B 40
C 60
D 5
Answer:
50% of a number means that we are taking half of a number. Since 11 is 50% of a number that means that we need to multiply 11 by 2 to get the original number. When we multiply 11 by 2 we get 22 and that is closest to option A which is 20.
Answer: Option A, [tex]20[/tex]
QUESTION 10
Which term best describes the expression 3x + 5?
O Quartic
O Quadratic
O Linear
O Cubic
