The statement "If f is differentiable at a, then f is continuous at a" is true, but "If f is continuous at a, then f is differentiable at a" is false.
Differentiability and continuity are two important concepts in calculus. A function is said to be continuous at a point if the values of the function approach a finite limit as the independent variable approaches the same value. In other words, there are no sudden jumps or breaks in the function at that point.
On the other hand, a function is said to be differentiable at a point if the limit of the rate of change of the function exists as the independent variable approaches the same value. This rate of change is called the derivative of the function and it is used to determine the slope of the function at that point.
So, it is true that if a function is differentiable at a point, then it is also continuous at that point. In other words, a continuous function must be differentiable. However, the reverse is not necessarily true. A function can reach a point where it is continuous but not differentiable. For example, the absolute value function is continuous everywhere but it is not differentiable at x = 0.
Therefore, we can say that if f is differentiable at a, then f is continuous at a. But if f is continuous at a, it does not imply that f is differentiable at a.
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It's pretty easy... Please help me ASAP and show your work. If I get 2 questions I will mark one person BRAINLIEST. So PLEAASEEEEEEEEEEE
Help on #4, #7, and #8
The given problems are solved:
What is an equation?An equation is a statement that asserts the equality of two expressions, the expressions are written one on each side of an '=' equal to sign.
Question 4:
To calculate angle A,
Consider tan function.
[tex]tan A=\frac{opposite side }{adjacent side}[/tex]
[tex]tan A=\frac{9.6}{7.2}[/tex]⇒ 1.34
A= 53.26°
Question 7:
In triangle PQR, ∠Q=80°, PQ=8 and QR= 6,
PR=?
Area of triangle = 1/2 *a*b*sinC
= 1/2*8*6*sin°80
=24*0.984
Area of triangle = 23.616 square units.
Question 8:
Consider the triangle JKL, here two sides and one angle is given,
we can use cosine law,
The equation set us is given by,
[tex]a^{2} = b^{2}+ c^{2} -2bcCos\alpha\\b ^{2} = a^{2}+c^{2} -2acCos\beta\\ c^{2} = b^{2} +a^{2} -2baCos\gamma[/tex]
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What type of graph is shown, and what is the growth factor?
linear function; growth factor of 2
linear function; growth factor of 4
exponential function; growth factor of 2
exponential function; growth factor of 4
The graph is a linear function with a growth factor of 4
What is Slope of Line?The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.
The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is
m=y₂-y₁/x₂-x₁
The graph has the shape of a straight line, so it is linear.
The graph starts at point (0, 0). It passes through point (1, 4).
slope = 4/1 = 4
The slope is 4. Growth factor is the same as slope for a linear function.
Hence, the graph is a linear function with a growth factor of 4
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1. Write in function notation a rule that moves a pre-image 4 units right and 7 units down.
A. f(x) (x + 4, y + 7)
B. f(x) (x + 4, y - 7)
C. f(x)(x-4, y + 7)
D. f(x) (x - 4, y - 7)
The function notation that moves a pre-image 4 units right and 7 units down is f(x) ⇒ (x + 4, y - 7)
What is translation?Translation is the movement of a point on the coordinate plane either left, right, up or down. Translation is a rigid transformation because it preserves the shape and size. Reflection and rotation are also rigid transformations.
If a point A(x, y) is translated a units right and b units down, then the new point is A'(x + a, y + b)
The function notation that moves a pre-image 4 units right and 7 units down is (x, y) ⇒ (x + 4, y - 7)
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Use the formula d = rt. Find r for t = 5h and d = 240 m.
The value of the variable r when t = 5h and d = 240m is 48m/h
What are algebraic expressions?Algebraic expressions are described as expressions that are composed of terms, variables, factors, coefficients and constants.
They are also made up of mathematical operations like subtraction. division, addition, multiplication, floor division, etc
From the information given, we have the equation;
d = rt
Make 'r' the subject of formula
r = d/t
Now, substitute the values into the equation, we get
r = 240/5
Divide the values
r = 48m/h
Hence, the value is 48m/h
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To indirectly measure the distance across a lake, Katherine makes use of a couple landmarks at points LL and MM. She measures KNKN, NLNL, and NONO as marked. Find the distance across the lake (LM)(LM), rounding your answer to the nearest hundredth of a meter.
The required distance across the lake (LM) is 205 m.
Similar triangles are those triangles that have similar properties,i.e. angles and proportionality of sides.
Here,
As from the figure,
Triangle KLM is similar to triangle KNO so,
KL/KN = LM/NO
180+70/180 = LM/147.6
LM = 205 m
Thus, the required distance across the lake (LM) is 205 m.
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Quadrilateral MNOP is similar to quadrilateral QRST. Find the measure of side RS.
Figures are not drawn to scale.
The measure of side RS is 7.6 units .
Similar quadrilaterals:
A quadrilateral's interior angles add up to 360 degrees. Two quadrilaterals are said to be identical when two adjacent sides have the same ratio and all three matching angles are equal (the fourth angle automatically becomes equal because the internal angle sum equals 360 degrees).
The corresponding angles of similar quadrilaterals should be equal in accordance with the similarity of quadrilaterals. Since we know that every angle in a square is 90 degrees, every matching angle in any two squares will be the same.
In the given question ,
Quadrilateral MNOP is similar to quadrilateral QRST .
[tex]\frac{ON}{NM} =\frac{RS}{RQ} \\\\\frac{2}{4} = \frac{RS}{15.2} \\\\\frac{1}{2} =\frac{RS}{15.2} \\\\RS=\frac{15.2}{2} \\\\RS=7.6units[/tex]
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on a linear graph, this is called the
In ΔNOP, p = 490 inches, m∠N=57° and m∠O=55°. Find the length of n, to the nearest inch.
The value of n is 443 inches when using the triangle rule and law of sine.
What is meant by Sine Rule?An oblique triangle's sides and angles are related according to the rule of sines (non-right triangle). The laws of sines and cosines are crucial trigonometrical formulas for "solving a triangle." The sine rule states that there is an equal relationship between a triangle's side lengths and the sine of each of its opposite angles.Triangles' side length to opposite angle ratios are governed by the law of sines. For all three sides and perpendicular angles, this ratio remains constant. So long as the necessary information is known, we may use the sine rule to determine the missing side or angle of any triangle.The formula is given as
a / sin A = b / sin B = c / sin C
In this triangle NOP, we can find the value of angle P
∠N + ∠O + ∠P = 180
57 + 55 + ∠P = 180 : Sum of angles in a triangle is equal 180 degrees
∠P = 180 - 112
∠P = 68
Using law of sine
p/sin P = n/sin N
490/sin68 = n / sin57
n = [(490 × sin 57) / sin 68]
n = 443 inches
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treena won a 7 day scholarship with $1000 to the pro shet basketball camp. rand trip travel exanse to the camp are $335 bh air or $125 by train. at the camp she much choose beteeen a week of individual instruction at $60 per dah or a week of group instruction at $40 per day. treenas food if she does not plan to spef money ofter then the scholarship ehat atr all choices of travel and induction plans that she could afford to make?
Answer:
To solve this problem, we need to calculate the cost of each travel and instruction option and then determine which options Treena can afford with her $1000 scholarship.
Option 1: Air travel and individual instruction
Travel cost: $335
Instruction cost: 7 x $60 = $420
Total cost: $335 + $420 = $755
Option 2: Air travel and group instruction
Travel cost: $335
Instruction cost: 7 x $40 = $280
Total cost: $335 + $280 = $615
Option 3: Train travel and individual instruction
Travel cost: $125
Instruction cost: 7 x $60 = $420
Total cost: $125 + $420 = $545
Option 4: Train travel and group instruction
Travel cost: $125
Instruction cost: 7 x $40 = $280
Total cost: $125 + $280 = $405
Now we need to determine which options Treena can afford with her $1000 scholarship. We'll start with the most expensive option and work our way down.
Option 1: $755
This option is too expensive for Treena's $1000 scholarship.
Option 2: $615
This option is also too expensive for Treena's $1000 scholarship.
Option 3: $545
This option is within Treena's budget, but she won't have any money left for food.
Option 4: $405
This option is also within Treena's budget, and she'll have $595 left for food.
Therefore, Treena's choices of travel and instruction plans that she could afford to make are:
- Train travel and group instruction with $595 left for food
- Train travel and individual instruction with no money left for food
Which of the following is the product of the rational expressions shown
below?
x+6 2-6
x+3 x-3
A. 22-12
B. 2-12
C. 2-366
9
OA.
OD. 12
The product of the rational expression shown above include the following: B. x² - 36/(x² - 9).
What is a rational expression?In Mathematics, a rational expression simply refers to a type of expression which is expressed as a fraction. This ultimately implies that, a rational expression is composed of two (2) main parts and these include the following:
NumeratorDenominatorIn this exercise, we would determine the product of the numerator for this rational expression and the product of the denominator for this rational expression as follows;
Product of numerator = (x + 6)(x - 6)
Product of numerator = x² + 6x - 6x - 36
Product of numerator = x² - 36
For the denominator, we have the following:
Product of denominator = (x + 3) × (x - 3)
Product of denominator = x² - 3x + 3x - 9
Product of denominator = x² - 9
Therefor, the product of the rational expression is given by x² - 36/(x² - 9).
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I can’t figure it out
Answer:
Step-by-step explanation:
x/y=(5/6)/(3/4) = 5/6*4/3 = 10/9
x/y=10/9
y/x=9/10
y=9/10 * x
Second option
Answer:
y=5/8x
Step-by-step explanation:
how to find the missing dimension of. a cone when the when the highest s 4.2 and the volume is 3.6 cubed
To find the missing dimension of a cone when the height is 4.2 and the volume is 3.6 cubed is to equate the general formula with the given information.
What is a cone?A cone is a three-dimensional geometric shape with a smooth and curving surface and a flat base, with an increase in the height radius of a cone decreasing to a certain point.
The volume of a cone is (1/3)πr²h.
The total surface area of a cone is πr(r + l).
The curved surface area is πrl.
To obtain the missing dimension of a cone we must equate the general formula for the volume of a cone to the given information.
Therefore, (1/3)πr²×4.2 = 3.6
(1/3)πr² = 0.86.
r² = 0.86/1.05.
r² = 0.82.
r = 0.9
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Please help asap thank you so much if you do
the relation has the domain of -4> x> 4 and the given relation is continuous for the given range.
What is the range or domain of a function?The range of values that we are permitted to enter into our function is known as the domain of a function. The x values for a function like f make up this set (x). A function's range is the collection of values it can take as input. After we enter an x value, the function outputs this sequence of values.
Given a graph,
The relation that defines the set of inputs to the set of outputs is called the functions. In function, each input in set X has exactly one output in set Y. Note: All functions are relations but all relations are not functions.
thus the given graph is a relation but not a function.
Given relation is continuous in the domain.
since the domain of the relation is The set which contains all the first elements of all the ordered pairs of relation R.
the given relation and has x values from -4 to 4 thus, the domain of the relations -4> x> 4.
Therefore, the relation has the domain of -4> x> 4 and the given relation is continuous for the given range.
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A. Find the probability of getting a five card hand that has three Jacks and two Queens. Round your answer to six decimal places.
b. A five card hand, where four of the cards are the same, i. E. Four Aces, or four Kings, etc. , is called getting a "four of a kind". What is the probability of getting a "four of a kind"? Round your answer to five decimal places
The probability of getting: A. a five-card hand that has three Jacks and two Queens is 0.000009, or 0.000923% and B. a "four of a kind" is 0.00024, or 0.02401%.
A. The probability of getting a five-card hand that has three Jacks and two Queens can be calculated by using the formula:
P(3 Jacks and 2 Queens) = C(4,3) x C(4,2) / C(52,5)
Where C(n,k) is the combination of n items taken k at a time.
= (4 x 6) / 2598960
= 24 / 2598960
= 0.00000923
Therefore, the probability of getting a five-card hand that has three Jacks and two Queens is 0.000009, or 0.000923% when rounded to six decimal places.
B. The probability of getting a "four of a kind" can be calculated by using the formula:
P(four of a kind) = (C(13,1) x C(4,4) x C(12,1) x C(4,1)) / C(52,5)
= (13 x 1 x 12 x 4) / 2598960
= 624 / 2598960
= 0.00024010
Therefore, the probability of getting a "four of a kind" is 0.00024, or 0.02401% when rounded to five decimal places.
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Convert 0.807 to a fraction in simplest form and a percent.
6y^2+25y+4 factor the expression
Answer: (6y + 1) (y + 4)
Step-by-step explanation: (6y + 1) (y + 4) (6y)(y) + (6y)(4) + (1)(y) + (1)(4) 6y^2 + 24y + 1y + 4 6y^2 + 25y + 4
Answer:
multiply 6×2 an you get 12
step:1 =4×12y⁴ ¹+25y¹ ¹
step:2=48y⁴ ¹ +25y¹ ¹
step:3=48y³+25y⁰
step:4=48y³+25×1
step:5=48y³+25
answer=48y³+25
Abbie has put $40 into her savings account. She has made 5 deposits of the same amount. How much was the amount of each deposit?
The amount of each deposit that Abbie made was $8
Calculate amount of depositThe amount of each deposit that Abbie made can be found by dividing the total amount she put into her savings account by the number of deposits she made.
In this case, we can use the formula: Amount of each deposit =
Total amount / Number of deposits
Substituting the given values into the formula, we get:
Amount of each deposit = $40 / 5
Amount of each deposit = $8
Therefore, the amount of each deposit that Abbie made was $8. In conclusion, if Abbie has put $40 into her savings account and has made 5 deposits of the same amount, then the amount of each deposit is $8.
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Given f(x)=-5x-3, find f-^1(x).
The inverse of the given function f(x) = -5x-3, is f⁻¹(x) = -x/5-3/5
What is an inverse function?An inverse function is a function that undoes the action of the another function. A function g is the inverse of a function f if whenever y=f(x) then x=g(y).
Given that a finction f(x) = -5x-3, we need to find the inverse of the function,
We can write,
y = -5x-3,
Interplace, x and y,
x = -5y-3
-5y = x+3
y = -(x+3)/5
y = -x/5-3/5
f⁻¹(x) = -x/5-3/5
Hence, the inverse of the given function f(x) = -5x-3, is f⁻¹(x) = -x/5-3/5
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Solve the following problems on numbers. Show your solution.
1. The sum of the two numbers is 91. The smaller number is 17 less than the larger number. What are the numbers? (2 points)
2. Twice the sum of a number and 8 is 24. Find the number. (1 point)
3. Find three consecutive integers such that the sum of the first and the third is 8 more than the second number. (3 points)
Answer:
Q1. The larger number is 54 and the smaller number is 37.
Q2. The number is 4.
Q3. The three numbers are 7, 8 and 9.
Step-by-step explanation:
Q1.
Let the larger number be x and the smaller number be y.
x+y= 91 (1)
x-y= 17 (2)
From (2) :
x= y+17 (3)
sub (3) into (1) :
y+17+y= 91
2y= 74
y= 37 (4)
sub (4) into (3) :
x= 37+17
= 54
Q2.
Let the unknown number be x.
2x+16= 24
2x= 8
x= 4
Q3.
Let the smallest number be x.
2nd no. : x+1
3rd no. : x+2
x+(x+2)= (x+1)+8
2x+2= x+9
x= 7
x+1= 8
x+2= 9
solve for v. 4v-18=6(v-6)
Answer:
v = 9
Step-by-step explanation:
4v - 18 = 6(v - 6)
1.) Distribute 6 into the parenthesis
4v-18 = 6v - 36
2.) Move 6v to the other side by subtracting from both sides
4v - 6v -18 = -36
3.) Move 18 to the other side by adding 18 on both sides
4v - 6v = 18 + -36
4.) Combine like terms
-2v = -18
5.) Divide -2 on both sides to get v by itself
v = 9
Answer:
the answer is 9
Step-by-step explanation:
4v-18=6(v-6)
4v-18=6v-36
4v-6v= -36+18
-2v= -18
divide both sides by -2
v=9
Write a ratio for each scenario using a colon(:).
1. Two cookies per student. __:__
2. The odds of the horse winning the race are 2 to 5. __:__
3. Colorado State University, there are 4 female students for every 3 male students. __:__
PLEASE BE ACCURATE!!! THANK YOU!!!
Answer:
2:1
2:5
4:3
Step-by-step explanation:
In these examples the comparison words "per", "to", "for every" is the colon.
Then just put the numbers in the same order they are in in the problem
1. Two cookies per student. __:__
Means 2 cookies per 1 student
We're writing the ratio
cookies:students
2:1
2. The odds of the horse winning the race are 2 to 5. __:__
The "to" is the colon
2:5
3. Colorado State University, there are 4 female students for every 3 male students. __:__
We are comparing
female:male
4:3
What is the integrated rate law for a 1st order reaction?
What is the symbol 1 to 10?
(Answer for 30 points) Penelope is collecting coins to raise money for a charity so far she collected
• twice as many dimes as quarters
• 4 times nickels than dimes, and
• three times as many pennies as quarters
If Penelope has collected 780 coins, how many of each type of coin she collected.
Penelope has, 52quarters, 104 dimes, 416 nickels, and 162 pennies.
What is a numerical expression?A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.
From the given information let, 'x' be the number of quarters,
Therefore, Dimes is 2x, Nickels is 8x and Pennies is 3x.
So, x + 2x + 8x + 4x = 780.
15x = 780.
x = 52, The number of quarters.
2×52 = 104, The number of Dimes.
8×52 = 416 Nickels.
3×52 = 162 Pennies.
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Find the median and mean of
the data. Which reflects the
best measure of the center?
X
X
8
X
X
X
X
X
9
XX
10
X
40
the mean and the median of the given data set are 16.75 and 9.5 respectively.
What is Mean?The sum of all values divided by the total number of values determines the mean (also known as the arithmetic mean, which differs from the geometric mean) of a dataset. The term "average" is frequently used to describe this measure of central tendency.
Given, A data set 8, 9, 10, 40
From the general formula of mean:
mean = (sum of observations) ÷ (total number of observations).
mean = (8 + 9 + 10+ 40)/4
mean = 67/4
mean = 16.75
Since the median is:
The median is the value that's exactly in the middle of a dataset when it is ordered. It's a measure of central tendency that separates the lowest 50% from the highest 50% of values.
Thus,
the median of the given data set = (9 +10)/2
the median of the given data set = 9.5
Therefore, the mean and the median of the given data set are 16.75 and 9.5 respectively.
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What is the surface area of a rectangular prism if the digits are 10ft, 1ft, and 5ft
The surface area of the rectangular prism is 130ft^2.
Determine surface areaThe surface area of a rectangular prism can be calculated by finding the area of each of the six faces and then adding them together.
The formula for the surface area of a rectangular prism is 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height.
In this case, the length is 10ft, the width is 1ft, and the height is 5ft. Plugging these values into the formula, we get:
Surface area = 2(10ft)(1ft) + 2(10ft)(5ft) + 2(1ft)(5ft)
Surface area = 20ft^2 + 100ft^2 + 10ft^2
Surface area = 130ft^2
Therefore, the surface area of the rectangular prism is 130ft^2.
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What is true of the triangle below?
Answer:
Step-by-step explanation:
where is the triangle
(Compound Interest and Geometric Sequences ) The equation, A equal to P quantity 1 plus 0.054 over 2 end quantity all raised to the power of 2 times t, represents the amount of money earned on a compound interest savings account with an annual interest rate of 5.4% compounded semiannually. If the initial investment is $3,000, determine the amount in the account after 15 years. Round the answer to the nearest hundredths place. $3,164.19 $6,671.67 $4,473.81 $14,532.47
The amount in the account after 15 years after round of to the nearest hundredth is $6,671.67, the correct option is B.
How to find the compound interest?If n is the number of times the interested is compounded each year, and 'r' is the rate of compound interest annually, then the final amount after 't' years would be:
[tex]a = p(1 + \dfrac{r}{n})^{nt}[/tex]
We are given that;
The amount earned is A equal to P quantity 1 plus 0.054 over 2 end quantity all raised to the power of 2 times t
Rate of compound interest= 5.4%
Investment= $3000
Time= 15years
Now,
A = 3000(1+0.054/2)^2*15
A = 3000(1+0.054/2)^30
A = $6,671.67
So, amount = principal + interest
Principal = $3,000.00
Interest = $3,671.67
=$6,671.67
Therefore, the compound interest will be $6,671.67
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PLEASE HELP ASAP‼️‼️Are the sides proportional?
Set up each ratio of sides and determine if they are equal
ratios. Show all of your work.
3.61 cm
R
44°
S
11.97 cm
9.71 cm
15
N
T
17.955 cm
14.565 cm
121°
M
L
5.415 cn
The side lengths form a proportional relationship, and the ratios are as follows:
17.955/11.97 = 14.565/9.71 = 5.415/3.61.
What are similar triangles?Similar triangles are triangles that share these two features presented as follows:
They must have Congruent angle measures.
They have Proportional side lengths.
Here, the triangles are similar, as they have the same angle measures.
17.955/11.97 = 14.565/9.71 = 5.415/3.61.
Thus, the side lengths form a proportional relationship, as the equivalent side have a ratio of 1.5.
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A brownie recipe calls for 1 cup of butter for every 2 cups of flour. Erika is using 8 cups of flour. How much butter should she use?