Quotient rule:
The Quotient Rule says that the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
The quotient rule is,
[tex]\frac{d}{dx}(\frac{u}{v})=\frac{vu^{\prime}-uv^{\prime}}{v^2}[/tex]Here, u and v are the functions of the same independent variable.
Example:
Let us take the problem,
[tex]y=\frac{\sin x}{\cos x}[/tex]Differentiating with respect to x using the quotient rule is,
[tex]\begin{gathered} \frac{dy}{dx}=\frac{\cos x\cdot\frac{d(\sin x)}{dx}-\sin x\cdot\frac{d(\cos x)}{dx}}{(\cos x)^2} \\ =\frac{\cos x(\cos x)-\sin x(-\sin x)}{\cos ^2x} \\ =\frac{\cos ^2x_{}+\sin ^2x}{\cos ^2x} \\ =\frac{1}{\cos ^2x} \\ =\sec ^2x \end{gathered}[/tex]A flower bed has the shape of a rectangle 5 yards long and 4 yards wide. What is its area in square feet?Be sure to include the correct unit in your answer.ftft 2ft 3
We would apply the formula for calculating the area of a rectangle which is expressed as
Area = length x width
From the information given,
length = 5
width = 4
By substituting these values into the formula,
Area = 5 x 4 = 20 yard^2
We would convert 20 yard^2 to ft^2
Recall,
1 yard^2 = 9 ft^2
Thus,
20 yard^2 = 20 x 9 = 180
The area is 180 ft^2
The roof of a building is in the shape of a square pyramid 28m on each side. If the slant height of the pyramid is 25m how much will roofing material cost at $8.59 per square m?
SOLUTION
Given the question in the question tab, the following are the solution steps to get the cost of the roofing material
Step 1: The square pyramid has 4 triangles. To get the cost, we first find the area of the 4 triangles
[tex]\begin{gathered} side=base=26m \\ \text{height}=25m \\ \text{Area}=\frac{1}{2}\times base\times height \\ \text{Area}=\frac{1}{2}\times26\times25 \\ \text{Area}=325m^2 \\ \text{Area of the 4 triangles}=4\times325=1300m^2 \end{gathered}[/tex]Step 2: Calculate the cost of the roofing material
[tex]\begin{gathered} 1m^2=8.59 \\ 1300m^2=x \\ By\text{ cross multiplication,} \\ x\times1=1300\times8.59 \\ x=11167 \end{gathered}[/tex]Hence, the cost of the roofing material for the building will be $11167
What is the area of this figure? 7 in 15 in 19 in 11 in 9 in 12 in 7 in Write your answer using decimals, if necessary, square inches
Question:
Solution:
Area of the vertical square:
A1 = b x h = 19 in x 7 in = 133 in^2
Area of the horizontal square:
A2 = b x h = 11 in x 4 in = 44 in^2
Area of the triangle:
[tex]A3\text{ = }\frac{b\text{ x h }}{2}\text{ = }\frac{11\text{ x 7}}{2}\text{ = }\frac{77}{2}\text{ = 38.5}[/tex]
what is the sum of -3x squared minus -7x + 9 and -5x squared + 6x minus 4
Answer:
The sum is;
[tex]-8x^2-x+5[/tex]Explanation:
We want to know the sum of;
[tex]-3x^2-7x+9\text{ and }-5x^2_{}+6x-4[/tex]Let Add and collect the like terms;
[tex]\begin{gathered} -3x^2-7x+9+(-5x^2_{}+6x-4) \\ =-3x^2-7x+9-5x^2_{}+6x-4 \\ =-3x^2-5x^2_{}-7x+6x+9-4 \\ =-8x^2-x+5 \end{gathered}[/tex]Therefore, the sum is;
[tex]-8x^2-x+5[/tex]Factorize the equations using the HCF method ,DOTS method , Trinomial method based on the equation
Answer:
To factorize the given equation:
[tex]8x^2-10x-3[/tex]Since the given equation is trinomial, we can use HCF method and trinomial method.
Dots method is only for binomial, it is not used for this equation.
By HCF method,
[tex]8x^2-10x-3=0[/tex][tex]8x^2-12x+2x-3=0[/tex][tex]4x(2x-3)+1(2x-3)=0[/tex]Taking (2x-3) as common we get,
[tex](2x-3)(4x+1)=0[/tex]we get,
[tex]\begin{gathered} 2x-3=0;4x+1=0 \\ \end{gathered}[/tex][tex]x=\frac{3}{2};x=\frac{1}{4}[/tex]The factors are,
[tex](2x-3)(4x+1)[/tex]The midpoint of AB is M(4, -2). If the coordinates of A are (3, -6), what are thecoordinates of B?
In general, given two points, the midpoint between those two points is given by the formula below
[tex]\begin{gathered} (x_1,y_1),(x_2,y_2) \\ \Rightarrow midpoint=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) \end{gathered}[/tex]Therefore, in our case, if B=(x,y),
[tex]\begin{gathered} (4,-2)=(\frac{3+x}{2},\frac{-6+y}{2}) \\ \Rightarrow4=\frac{3+x}{2} \\ and \\ -2=\frac{-6+y}{2} \\ \Rightarrow8=3+x \\ and \\ -4=-6+y \\ \Rightarrow x=5 \\ and \\ y=2 \end{gathered}[/tex][tex]\Rightarrow B=(5,2)[/tex]Therefore, the answer is B=(5,2)f(x) = = between x=4 and x = 7 using the "midpoint rule" with two rectangles of equal width.
EXPLANATION
Since we have the function f(x)= 1/x
[tex]\int _a^bf\left(x\right)dx\:\approx \Delta \:x\left(f\left(\frac{x_0+x_1}{2}\right)+f\left(\frac{x_1+x_2}{2}\right)+f\left(\frac{x_2+x_3}{2}\right)+...+f\left(\frac{x_{n-1}+x_n}{2}\right)\right)[/tex][tex]\mathrm{where}\:\Delta \:x\:=\:\frac{b-a}{n}[/tex]Given a=4, b=7, n=2
[tex]Δx=\frac{7-4}{2}=\frac{3}{2}[/tex]Divide the interval, 4<=x<=7 into 2 subintervals:
[tex]x_0=4,\:x_1=\frac{11}{2},\:x_2=7[/tex][tex]=\frac{3}{2}\left(f\left(\frac{x_0+x_1}{2}\right)+f\left(\frac{x_1+x_2}{2}\right)\right)[/tex]Calculate the subintervals:
[tex]f\left(\frac{x_0+x_1}{2}\right)=f\left(\frac{4+\frac{11}{2}}{2}\right)[/tex][tex]=f\left(\frac{19}{4}\right)[/tex][tex]=\frac{1}{\frac{19}{4}}[/tex][tex]=\frac{4}{19}[/tex][tex]f\left(\frac{x_1+x_2}{2}\right)=f\left(\frac{\frac{11}{2}+7}{2}\right)[/tex][tex]=\frac{4}{25}[/tex][tex]=\frac{3}{2}\left(\frac{4}{19}+\frac{4}{25}\right)[/tex]Applying the distributive property and adding fractions:
[tex]=\frac{264}{475}[/tex]In conclusion, the solution is 264/475
mr. Williams wants to get better at basketball last year he made 150 free throws in a row this year has been practicing made 273 what is his percentage difference?
We are asked to find the percentage difference. This can be seen below;
Explanation:
We will use the formula below;
[tex]\text{Percentage difference = }\frac{New\text{ throws -old throws}}{\text{old throws}}\times100[/tex]Therefore
[tex]\begin{gathered} \text{Percentage difference = }\frac{273-150}{150}\times100 \\ =\frac{123}{150}\times100 \\ =82\text{\%} \end{gathered}[/tex]Answer: 82%
I need a tape diagram for the equation: 2 + 6x =32and another tape diagram for the equation: 7x + 1 = 15
Explanation:
2 + 6x =32
tape diagram:
Solve the following system using the substitution method.y = -5x - 17-3x - 3y = 3
Let:
[tex]\begin{gathered} y=-5x-17_{\text{ }}(1) \\ -3x-3y=3_{\text{ }}(2) \\ \end{gathered}[/tex]Replace (1) into (2):
[tex]undefined[/tex]In the equation y= 2x + 7, the slope is 7. TrueFalse
In a linear equation like y=2x+7 the slope represents
Answer:
False
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = 2x + 7 ← is in slope- intercept form
with slope m = 2 and y- intercept c = 7
I need help with my math
No solution.
1) Examining those lines we can notice that these lines do not intercept each other in any point.
2) So we have two parallel lines.
3) Hence, we can state that this Linear System has no solution.
ABC is a right triangle and the measure of angle B is 49 .What is the measure of C?
ABC is a right triangle, and one of the angles is 49, so:
Using the triangle sum theorem:
[tex]\begin{gathered} A+B+C=180 \\ 90+49+C=180 \\ C=180-90-49 \\ C=41^{\circ} \end{gathered}[/tex]Stacy works at a clothing store. Her weekly salary is $200. In addition to her say the earns 5% of the amount of her sales each week. What is the total amount Stacy should earn if she sells $1000 worth of clothing in 1 week?
$250
1) Gathering the data
Weekly Salary = $200
+ 5%
Total sales in one week = $1000
2) We can find the earnings of Stacy calculating this. Let's convert 5% to 0.05
Earnings = 200 + 1000 x (0.05)
E= 200 +50
E= 250
3) So Stacy should earn in 1 week $250 (salary +sales percentage).
I think I understand the problem but I just wanted to check if I got it right.
let's solve the equation system:
since we can equalize both equation of y, we obtain:
[tex]4x-1=2x-5[/tex]now we need to solve this equation for x:
[tex]4x-2x=-5+1[/tex][tex]2x=-4\Rightarrow x=-2[/tex]now that we found x, we just need to replace it in one of the two equations of y, to find it:
[tex]y=4x-1=4(-2)-1=-9[/tex]then the final answer is y=-9 and x=-2
thank you too for your question, have a beautiful day.
The restaurant bill for Annette is $55.82. She adds an 18% tip for the waiter. What is the total cost of Annette's meal rounded to the nearest cent?
Given:
Resturant bill = $55.82
She adds 18% tip of bill:
[tex]0.18\text{ }\ast\text{ 55.82 = \$}10.0476[/tex]Therefore, the total cost of the meal is:
Suppose you flip four coins and you want to determine the probability ofexactly three of these coins flipping a head..If flipping three heads is afavorable compound event, what ratio can be used to determine thisprobability?
Ok, so:
From the problem, we know that there are two possibilities when we are flipping a coin. It could be head, or stamp.
So, we notice that flipping three heads is a favorable compound event, and the probability of any situation has the following formula:
P = (favorable compound event) / (All possible events)
If we have four coins, we notice that there are four heads and four stamps. So, if we flip the four coins, we would get 8 possible events.
Then, if 3 coins flipping a head is a favorable compound event and 8 are all the possible events, probability will be:
P = 3/8.
Express the interval in set-builder notation and graph the interval on a number online.(-Infinity,4.1)
Given:
Interval is:
[tex](-\infty,4.1)[/tex]Find-:
Interval on a number line.
Explanation-:
The number line is:
The interval is:
[tex](-\infty,4.1)[/tex]In number line graph is:
what is negative 1\6 times 2 and 3/4?
When we multiply a number by a fractional number, we can express it as the multiplication of the numeratorand the fraction divided by the ddenominatorlike this:
[tex]a\times\frac{b}{c}=\frac{a\times b}{c}[/tex]And with the multiplication of fractional numbers we can do:
[tex]\frac{a}{d}\times\frac{b}{c}=\frac{a\times b}{d\times c}[/tex]In this case, we have -1/6 times 2 times 3/4:
[tex]-\frac{1}{6}\times2\times\frac{3}{4}=-\frac{1\times2\times3}{6\times4}=-\frac{6}{6\times4}=-\frac{6}{6}\times\frac{1}{4}=-1\times\frac{1}{4}=-\frac{1}{4}[/tex]Then, the answer is -1/4
Scores on the GRE (Graduate Record Examination)are normally distributed with a mean of 598 and astandard deviation of 140. Use the 68-95-99.7 Ruleto find the percentage of people taking the test whoscore between 318 and 878.The percentage of people taking the test who scorebetween 318 and 878 is _ %
SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Describe the Emperical rule
The Empirical Rule(68-95-99.7 Rule) states that, for a normally distributed random variable:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
STEP 2: Write the given details
[tex]\begin{gathered} mean(\mu)=598 \\ standard\text{ }deviation(\sigma)=140 \end{gathered}[/tex]STEP 3: Solve the question
In this problem, we have that:
[tex]\begin{gathered} 318=598-280=598-2(\sigma)=\mu-2\sigma \\ 878=598+280=598+2\sigma=\mu+2\sigma \end{gathered}[/tex]So 318 is 2 standard deviations below the mean.
So 878 is 2 standard deviations above the mean.
The EMPIRICAL RULE (or the 68-95-99.7 Rule) says that about 68% of the data is between 1 standard deviation below the mean and 1 standard deviation above the mean. Below the shaded part is about 68% of the area between the normal curve and the z-axis.
The EMPIRICAL RULE (or the 68-95-99.7 Rule) also says that about 95% of the data is between 2 standard deviations below the mean and 2 standard deviations above the mean. Below the shaded part is about 95% of the area between the normal curve and the z-axis.
Going by this descriptions above, this means that:
The percentage of people taking the test who score between 318 and 878 is 95%
Graph the line with slope 3 passing through the point (2,2)
To answer this question we will use the following slope-point formula for the equation of a line:
[tex]y-y_1=m(x-x_1).[/tex]Therefore, the equation of the line that has a slope of 3 and passes through (2,2) is:
[tex]y-2=3(x-2).[/tex]Simplifying the above result we get:
[tex]y-2=3x-6.[/tex]Adding 2 to the above equation we get:
[tex]\begin{gathered} y-2+2=3x-6+2, \\ y=3x-4. \end{gathered}[/tex]From the last equation, we get that the y-intercept of the line is (0,-4).
Since the graph of the line also passes through (2,2) we get that its graph is:
Answer:
If the drinks portion of a bill costs $18 and the total bill was $30 what percentage of the total was the cost of the drinks?
we have to divide 18 by 30 and then multiply it by 100
[tex]\frac{18}{30}\cdot100=60\text{ \%}[/tex]4th grade math question Joey walk 333 ft how many yards did Joey walk? At 5 yds vs 15 ft
111 yards
Explanation
Step 1
to solve this we need to know the equivalence
[tex]\begin{gathered} 3\text{ ft= 1 yd} \\ 3\text{ ft}\leftrightarrow1\text{ yd} \end{gathered}[/tex]Now, we can apply a rule of three
let x represents the number of yards in 333 ft, so
so
[tex]\begin{gathered} if \\ 3\text{ ft}\rightarrow\text{ 1 yd} \\ \text{then} \\ 33\text{ ft}\rightarrow x \end{gathered}[/tex]as the ratio is the same, we have a proportion
[tex]\begin{gathered} \frac{3\text{ ft}}{1\text{ yd}}=\frac{33\text{ ft}}{x\text{ yd}} \\ \frac{3}{1}=\frac{33}{x} \end{gathered}[/tex]Step 2
now, solve for x
[tex]\begin{gathered} \frac{3}{1}=\frac{333}{x} \\ \text{cross multiply} \\ 3\cdot x=333\cdot1 \\ 3x=333 \\ \text{divide both } \\ \frac{3x}{3}=\frac{333}{3} \\ x=111 \end{gathered}[/tex]so, the answer is 111 yards
I hope this helps you
Find the side length of a cube with a volume of 531 yd^3If necessary, round your answer to the nearest tenth.
The volume of a cube is given by the following formula:
[tex]V=s^3[/tex]Where V is the volume and s is the side length.
Given the volume, to find the side length find the cubic root of the volume:
[tex]\begin{gathered} \sqrt[3]{V}=s \\ s=\sqrt[3]{531} \\ s=8.1 \end{gathered}[/tex]The side length of the cube is 8.1 yd.
Exponent Rule of algebra Please help meWhat is the order of this problem in the Picture1. Quotient rule2. Negative exponents rule 3. Exponent power rule4. Product rule5. Combining like termsIn order
That is the expression given in the exercise.
The procedure is:
1. Apply the Negative exponent rule:
[tex]=(\frac{6x^{-2}yz^{-5}}{-3x^{-6}y^{-1}z^{-2}})^2-(\frac{3x^3y^2}{z^2})^3\cdot\frac{1}{xy^2}[/tex]2. Apply the Expanded power rule:
[tex]=\frac{36x^{-4}y^2z^{(-10)}}{9x^{(-12)}y^{-2}z^{-4}}^{}-\frac{27x^9y^6}{z^6}^{}\cdot\frac{1}{xy^2}[/tex]3. Apply the Product rule :
[tex]=\frac{36x^{-4}y^2z^{(-10)}}{9x^{(-12)}y^{-2}z^{-4}}^{}-\frac{27x^9y^6}{z^6xy^2}[/tex]4. Apply the Quotient rule:
[tex]=\frac{4x^8y^4^{}}{^{}z^6}^{}-\frac{27x^8y^4}{z^6^{}}[/tex]5. Combining like terms, you get:
[tex]=\frac{4x^8y^4-27x^8y^4}{^{}z^6}^{}=-\frac{23x^8y^4}{z^6}[/tex]The answer is:
1. Negative exponent rule.
2. Expanded power rule.
3. Product rule.
4. Quotient rule.
5. Combining like terms.
Please explain this problem and How to solve this problem please explain step by step. Thanks
Determine if the lower bound theorem identifies -7 as a lower bound for the real zeros of
f (x) = 5x3 - 2x2 - 5x - 2
A Yes
B NO
=======================================================
Explanation:
Let c < 0 be a test root of f(x). The key here is that c must be negative.
If we use this value of c in the synthetic division table, and we get the third row alternating in signs, then we have proven c is a lower bound of the roots of f(x). This is what the lower bound theorem tells us.
In this case, c = -7 fits the description of c < 0.
Refer to the diagram below to see the synthetic division. The third row of values is: 5, -37, 254, and -1780
The terms alternate in sign which is sufficient evidence to conclude -7 is a lower bound for the real zeros of f(x).
What this means is that we don't need to look lower than -7 since we know -7 is like a "floor" of sorts. It's not the highest lower bound as there may be higher floor values.
-----------
For more information, search out [tex]\text{``lower bound theorem'' algebra}[/tex] where the quotes are needed. One of the results is from the West Texas A&M University (WTAMU) Virtual Math Lab, which may be handy.
A space shuttle is made usinga scale of 1 inch : 12 feet. Ifthe actual space shuttle has aheight of 84 feet, then themodel is inches tall.
A space shuttle is made using
a scale of 1 inch : 12 feet. If
the actual space shuttle has a
height of 84 feet, then the
model is inches tall.
we know that
the scale is 1 inch : 12 feet.
so
using proportion
1/12=x/84
solve for x
x=84/12
x=7 in
therefore
the answer is 7 inchesPart 2
subtract: (4b-8b+11) -(-2b -4b +5). There is a small 2 above the 4b in the first equation. There is a small 2 above the b in the second equation.
We are asked to solve the following subtraction:
[tex]\mleft(4b^2-8b+11\mright)-\mleft(-2b^2-4b+5\mright)[/tex]First we will change the sign inside the second parenthesis:
[tex]\mleft(4b^2-8b+11\mright)+\mleft(2b^2+4b-5\mright)[/tex]Now we will associate like terms:
[tex](4b^2+2b^2)+(-8b+4b)+(11-5)[/tex]Now we add the associated terms;:
[tex]6b^2-4b+6[/tex]And thus we get the answer.
You just got a really nice camera for your birthday. Your friends camera can take 20 pictures per minute. Your new camera can take 35 pictures per minute. If you both took pictures for 10 minutes, how many more pictures would your camera take?
If the new camera can take 35 pictures per minute and the other cameras can take 20 pictures per minute, the difference of pictures per minute is:
[tex]35-20=15[/tex]So the new camera can take 15 more pictures per minute. Then, in 10 minutes, the amount of more pictures the new camera has is:
[tex]15\cdot10=150[/tex]So in 10 minutes the new camera has taken 150 more pictures.
Consider the equation. Determine whether the graph of the equation is wider or narrower than the graph of Y
Both equations are symmetrical when related to the y-axis and have a vertex at (0,1). But, since coefficient 1/2 make the second equation grows slower, we can conclude that the graph of the first equation is narrower than the graph of the second equation.