Really need to someone to break this down so I can understand it
(a) Find the slope of the curve y= x^2 - 2x - 3 at the point P(2, -3) by finding the limit of the secant slopes through point P.
(b) Find an equation of the tangent line to the curve at P(2, -3)
Answers
Answer:
Part A)
The slope is two.
Part B)
[tex]\displaystyle y = 2x - 7[/tex]
Step-by-step explanation:
Part A)
We want to find the slope of the curve:
[tex]\displaystyle y = x^2 - 2x - 3[/tex]
At the point P(2, -3) by using the limit of the secant slopes through point P.
To find the limit of the secant slopes, we can use the difference quotient. Recall that:
[tex]\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}[/tex]
Since we want to find the slope of the curve at P(2, -3), x = 2.
Substitute:
[tex]\displaystyle f'(2) = \lim_{h \to 0} \frac{f(2 + h) - f(2)}{h}[/tex]
Simplify. Note that f(2) = -3. Hence:
[tex]\displaystyle \begin{aligned} f'(2) &= \lim_{h\to 0} \frac{\left[(2+h)^2 - 2(2+h) - 3\right] - \left[-3\right]}{h} \\ \\ &=\lim_{h \to 0}\frac{(4 + 4h + h^2)+(-4-2h)+(0)}{h} \\ \\ &= \lim_{h\to 0} \frac{h^2+2h}{h}\\ \\&=\lim_{h\to 0} h + 2 \\ \\ &= (0) + 2 \\ &= 2\end{aligned}[/tex]
(Note: I evaluated the limit using direct substitution.)
Hence, the slope of the curve at the point P(2, -3) is two.
Part B)
Since the slope of the curve at point P is two, the slope of the tangent line is also two.
And since we know it passes through the point (2, -3), we can consider using the point-slope form:
[tex]\displaystyle y - y_1 = m(x-x_1)[/tex]
Substitute. m = 2. Therefore, our equation is:
[tex]\displaystyle y + 3 = 2(x-2)[/tex]
We can rewrite this into slope-intercept if desired:
[tex]\displaystyle y = 2x - 7[/tex]
We can verify this by graphing. This is shown below:
Related Questions
help me out with this!
Answers
Answer:
k = 70
Step-by-step explanation:
Since we have been given the value of x to be -ve5 (-5),
we multiply 6 and -5( value of x). Which gives;
-30.
Then add the answer (-30) to 100.
-
K = -30 + 100
= 100 - 30
=70
Note;
(+ * + = +)
( + * - = -)
(- * - = +)
I hope this helps you.
Answer:
[tex]k=70[/tex]
Step-by-step explanation:
[tex]k=6x+100[/tex]
[tex]x=-5[/tex]
[tex]k=6(-5)+100[/tex]
Remove parentheses:- [tex]\left(-a\right)=-a[/tex]
[tex]k=-6\times 5+100[/tex]
Multiply [tex]6\times5= 30[/tex]
[tex]k=-30+100[/tex]
Add [tex]-30+100=70[/tex]
[tex]k=70[/tex]
OAmalOHopeO
Write the nth term of the following sequence in terms of the first term of the sequence. PLEASE HELP DUE IN 6 DAYS!!!!!!!
a, a + 4, a +8,...
Answers
Answer:
a + 4(n-1)
Step-by-step explanation:
The first term is a+4, the second term is a+8, the third term is a+12, etc.
A common theme that we can notice is that we add 4 for each term. For example, the second term (t₂) equals t₁+4, the third term equals t₂+4 (as a+4+4 = a+8), and so on. Another way of writing this is that we multiply 4 by (term number - 1) and add that to a. We can write this as
4 * (n-1) + a
WILL MARK BRAINIEST IF FAST!
Kane's Furniture Store advertised a table at a 11% discount. The original selling price was $111, and the sale price was $100. Was the sale price consistent with the ad? Explain.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
O A. No, the sale price was s higher than it should have been
(Round to the nearest cent.)
OB. No, the sale price was $ lower than it should have been
(Round to the nearest cent.)
OC. Yes, the sale price was consistent with the ad, because it represents a 11% discount.
va
Vi
(11)
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Click to select and enter your answer(s) and then click Check Answer.
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Answers
Answer:
A. No, the sale price was 1.21 higher than it should have been
Step-by-step explanation:
First find the discount
111 * 11%
111 * .11
12.21
Subtract this from 111
111-12.21
98.79
The sale price should have been 98.79
100-98.79
1.21
The difference of two numbers is 16, and their sum is 20. What are the two numbers?
Answers
Answer:
18 and 2
Step-by-step explanation:
18-2=16
18+2=20
Hi ;-)
[tex]\{x-y=16\\+\{x+y=20\\--------\\2x = 36 \ \ /:2\\\boxed{x=18}\\18+y=20 \ \ /-18\\\boxed{y=2}[/tex]
These numbers are 18 and 2.
In Evan's math class, there are 7 boys and 10 girls. Which of the following is the ratio of boys to girls in Evan's math class?
Answers
Answer:
7/17
Step-by-step explanation:
write rhis number in standard notation. 1.342×10^-7
Answers
Answer:
0.0000001342
Step-by-step explanation:
Help me pls!!!!!!!!!!!
Answers
Answer:
.01 times
Step-by-step explanation:
15) A small rectangular block of land has a perimeter of 100 m and an area of 225 m². Find the dimensions of the block of land.
Answers
Answer:
45 and 5 are the dimensions of the block of land.
Step-by-step explanation:
2(x + y) = 100, x + y = 100/2, y = (50 - x)
xy = 225
x (50 — x) = 225
0 = x^2 — 50x + 225
0 = (x — 45) (x -5)
45 and 5 are the roots of the above polynomial, thus they are the sides or dimensions of the block of land.
Simplify an expression for the perimeter of the rectangle.
W = 24.6x - 10.4
L = 7x + 3.1
Answers
The expression for the perimeter of the rectangle is 63.2x - 14.6.
Given that,
Width = W = 24.6x - 10.4Length = L = 7x + 3.1We Know that
Perimeter of the rectangle is
[tex]= 2(l+w)\\\\= 2 (7x + 3.1 + 24.6x - 10.4)\\\\= 2(31.6x - 7.3)[/tex]
= 63.2x - 14.6
Here the above formula should be applied for measuring the perimeter of the rectangle.
Therefore we can conclude that The expression for the perimeter of the rectangle is 63.2x - 14.6.
Learn more about the rectangle here: brainly.com/question/16167300
Answer:
24.3
Step-by-step explanation:
First you solve the sides:
24.6-10.4 =14.2
7+3.1=10.1
After that ,you will add because perimeter is adding up the side lengths of various shapes.
14.2+10.1=24.3
Expanded form for 4082 × 1/10=
Answers
Answer:
4000+0+80+2×0.1
Step-by-step explanation:
after expanding, /means division so you diving
PLZ HELP ASAP I NEED PLZZZ!!!!!!
Fill in the blank with the correct response. The slope of the graph of y = -2/3x is __.
Answers
Answer:
Solution given:
Equation of line is:
y=-⅔x
comparing above equation with
y=mx+c
we get
m=-⅔
So
slope is -⅔Answer:
slope is -2/3
Step-by-step explanation:
I. It's linear equation
y = mx + b
when y is output, x is input, m is slope and b is addition.
The statment above is y = -2/3x + 0
So -2/3 is slope or m.value
Hope that help :D
Can someone help me please :00
Answers
Answer:
according to the dark spots which shows where each number goes to, it would be D.-2
GENTE PORFAVOR ME AJUDA
Answers
You have asked for help without actually providing what you need help on.
Solve for x where 2/x=2exponent x
Answers
Answer:
x = 1
Step-by-step explanation:
2/x = 2^(x)
x = 1
hope that helps
please, solve for x!
Answers
Answer:
15
Step-by-step explanation:
x*x = 9*9 + 12*12
--> x = 15
Consider the points below. P(â1, 4, 1), Q(0, 6, 2), R(4, 3, â1)
a. Find a nonzero vector orthogonal to the plane through the points P, Q, and R.
b. Find the area of the triangle PQR.
Answers
Answer:
B. Find the area of the triangle PQR.
Which set of ordered pairs represents a function?
{(3,6), (3, 1), (9,0), (2, 2)}
{(2,-5), (-3, 6), (-1, -8), (-3,-4)}
{(4, -7), (-2, -1), (-7,4), (9, -1)}
{(3,8), (-2,-9), (1,4), (1,9)}
Answers
Answer:
{(4, -7), (-2, -1), (-7,4), (9, -1)}
Step-by-step explanation:
Every other set of ordered pairs has an x-value that has more than one y-value. This is against the rules of a function. Therefore, {(4, -7), (-2, -1), (-7,4), (9, -1)} represents a function.
lim x ->0 ln(1+5x)/2x
Answers
answer
−>0(1+5)/2
hope it helps
Is 0.8 equivalent to a. 8/12, b. 5/8, c. 2/3, d. 7/50, e. 4/5, f. 5/3, g. 19/25, h. 1/3, i. 2/9, j. 1/33, k. 4/9
Answers
Answer:
g. 19/25
Step-by-step explanation:
19/25 = .76, or approximately .8 -- none of the other options come as close as option g.
11. Find the solutions of x^2-x-30 = 0.
Please give a walkthrough/step by step I am trying to learn how to do it!
Answers
To find the solutions of quadratic equation, there are two ways to do which are:
FactorizeQuadratic FormulaStep 1
- Factor the expression.
To factor the expression, refer below:
[tex] \displaystyle \large{ (x - a)(x - b) = {x}^{2} - bx - ax + ab}[/tex]
For bx and ax, both can be common-factored. Therefore
[tex] \displaystyle \large{ (x - a)(x - b) = {x}^{2} - (b + a)x + ab}[/tex]
From the above, we conclude that:
The middle term is b+aThe last term is a×bThus, we have to find two numbers that satisfy a+b and a×bFrom the expression, 30 comes from 5×6 and when 5-6 = -1. Therefore, a can be 5 and b can be 6.
[tex] \displaystyle \large{{x}^{2} - x - 30 = (x + 5)(x - 6)}[/tex]
Because in the middle term, it is -x which is negative, we have to let the highest number become negative.
From the factored expression:
The middle term = 5x + (-6x) = -xThe last term = 5 × (-6) = -30Then we replace the standard equation with factored form.
[tex] \displaystyle \large{ (x + 5)(x - 6) = 0}[/tex]
For this part, we solve like a linear equation where we isolate x. Just think you are solving two linear equations!
Hence
[tex] \displaystyle \large{ x = - 5, 6}[/tex]
Therefore, the solutions are x = -5, 6.
If you drive 565 miles a day, how far will you have gone at the end of a week?
Answers
If you drive 565 miles a day for 7 days do 565 times 7!
Vito avido is a sales representative for a company that sells roofing material. he receives a graduated commission of 5 percent on the first $5,000 of sales, 6.5 percent on the next $15,000, and a 7 percent on sales over $20,000. What is his commission on $23,458 in sales?
Answers
Hence commission will be 7%
[tex]\\ \rm\longmapsto 23458\times \dfrac{7}{100}[/tex]
[tex]\\ \rm\longmapsto \dfrac{164206}{100}[/tex]
[tex]\\ \rm\longmapsto 1642.06[/tex]
Vito Avido's commission on $23,458 in sales is $1466.06.
Here, we have to calculate Vito Avido's commission on $23,458 in sales, we'll break down the sales amount into different ranges and apply the respective commission rates for each range.
For the first $5,000 of sales: 5% commission
Commission = 5% of $5,000 = 0.05 * $5,000 = $250
For the next $15,000 of sales: 6.5% commission
Commission = 6.5% of $15,000 = 0.065 * $15,000 = $975
For the remaining amount over $20,000: 7% commission
Sales above $20,000 = $23,458 - $20,000 = $3,458
Commission = 7% of $3,458 = 0.07 * $3,458 = $241.06
Now, add up the commissions from each range:
Total Commission = $250 + $975 + $241.06 = $1466.06
Vito Avido's commission on $23,458 in sales is $1466.06.
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Cecelia receives two credit card offers in the mail. Each has a 0% introductory APR. After the introductory period, the first card has an APR of 19.99% and the second card has an APR of 24.99%. If each card has a late fee of $35 and an annual membership fee of $25, which card should she choose and why?
Answers
Answer:
1. We are told they have similar APR (starting) and Annual fee of 25 and similar late fee of 35
Then we must consider the APR after the initial value 19.999% < 24.99%Hence she should price the first card with a 19.99% APR
You and your family are planning a vacation of a lifetime around the world that will cost $25,000. You borrowed this amount from your bank with an annual interest rate of 10.5%. You decided to pay it back in 5 years. What is the interest that will accrue on this loan?
$3,125.00
$131,250
$13,125.00
$312.50
Answers
Answer:
$13,125
Step-by-step explanation:
The value of the interest that will accrue on this loan will be;
⇒ $13,125
What is Simple interest?A quick and easy method of calculating the interest charge on a loan is called a Simple interest.
Given that;
You and your family are planning a vacation of a lifetime around the world that will cost $25,000.
And, You borrowed this amount from your bank with an annual interest rate of 10.5%.
Now,
Since, You borrowed this amount from your bank with an annual interest rate of 10.5%.
Hence, The total interest amount after five years is,
⇒ 5 × (10.5% of $25,000)
⇒ 5 × (10.5/100 × 25,000)
⇒ 5 × 2,625
⇒ $13,125
Thus, The total interest amount after five years = $13,125
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Brent needs to paint a logo made using two right triangles. The dimensions of the logo are shown below. What is the difference between the area of the large triangle and the area of the small triangle?
Two right triangles are attached along their longest side. The smaller triangle has height 6 cm and base 2 cm. The height of the larger triangle is 9 cm and its base is 5 cm.
4.5 cm2
8.5 cm2
16.5 cm2
33.0 cm2
Answers
9514 1404 393
Answer:
(c) 16.5 cm^2
Step-by-step explanation:
The area of each triangle is given by the formula ...
A = 1/2bh
Smaller triangle area = (1/2)(2 cm)(6 cm) = 6 cm^2
Larger triangle area = (1/2)(5 cm)(9 cm) = 22.5 cm^2
The difference in area is ...
22.5 cm^2 -6 cm^2 = 16.5 cm^2
Three times the sum of a number and 18 is at least-30
Answers
Answer:
the sum of the number is 16 and it is al least 20 points
Write 96% as a fraction.
Answers
Answer:
Directly, it would be 96/100. However, we can reduce it to become 48/50 = 24/25.
(underlined is simplest answer)
24/25
Step-by-step explanation:
96%
=96/100
=24/25
answer
Select the matrix multiplication that transforms vector A to
vector B
Answers
Answer:
Option C.
Step-by-step explanation:
Notice that only option B and C involves the vectors in transformation
[tex]A=\begin{bmatrix}4\\2\end{bmatrix}, B=\begin{bmatrix}-8\\-4\end{bmatrix}[/tex]
However, option B is not a defined multification since the dimension of both matrices are not matching. Vector A is 2x1 but the second matrix is 2x2.
Show how find 42 x 27 using an area number
Answers
Answer:
First break it down
42= 40 & 2
27= 20 & 7
Then times 40 & 20 to get 800
20 x 2 to get 40
40 x 7 to get 280
7 x 2 to get 14
Now add them all together.
800 40 1080
+280 +14 + 54
---------- ------- ---------
1080 + 54 = 1134
If point (-3,1) is in the function p(x), which of the following points will be in the function p-|(2)?
PLZ HELP I NEED IT
Answers
Answer:
C
Step-by-step explanation:
for p- prime you just swipe your co-ordinates ...n when swiping them don't change your sign
The point that will be in the function p⁻¹(x) is (1,-3), the correct option is C.
What is a Function?A function is a law that relates a dependent and an independent variable.
The function p(x) has point (-3, 1 )
An inverse of the function is obtained by interchanging the value of y and x, then solving the equation for y in terms of x
The point that will be in the inverse function of (x,y) is (y,x)
(-3,1) -----> ( 1,-3)
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