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3 years ago

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Finding an Equation of a Tangent Line In Exercise, find an equation of the tangent line to the graph of the function at the give

n point.
y = e^2x^2, (1, e^2)

1 answer:

Aloiza [94]3 years ago

4 0

Answer:

$y = 4e^2x - 3e^2$

Step-by-step explanation:

$f(x) = e^{2x^2}$

lets calculate the derivate of f using the chain rule:

$f'(x) = e^{2x^2}* (2x^2)' = e^{2x^2}*4x = 4e^{2x^2}x$

we have that f'(1) = 4e^2, hence the equation is

$y = f(1) + f'(1) (x-1) = e^2 + 4e^2(x-1)$

or, equivalently,

$y = 4e^2x - 3e^2$

You might be interested in

Triangles PQR and XYZ are similar triangles. Which are the other two angle measures of triangle XYZ? 35° 70° 75° 95° 105°

andrezito [222]

75 degrees. Hope that helps and if it doesn't sorry

7 0

4 years ago

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Write a coordinate proof for the following statement: Any triangle ABC formed so that vertex C is on the perpendicular bisector

AnnyKZ [126]

Answer:

Answer is contained in explanation.

Step-by-step explanation:

Description of visual:

I started with the first picture. This is a picture of triangle ABC.

Now I'm going to draw a line segment from vertex C such that it is a perpendicular bisector of AB.

Proof:

CM is a perpendicular bisectors of AB is a given.

From this we can concluded by definition of perpendicular angles that angle AMC and angle BMC are right angles.

Since angles AMC and BMC are right angles, then they are congruent to each other.

By the definition of bisector and since CM bisects AB, then AM is congruent to MB.

By the reflexive property, we have that CM is congruent to CM.

We can conclude the two triangles, triangle CMA and CMB, are congruent by SAS Postulate.

Since triangles CMA and CMB are congruent, we can conclude that their corresponding parts are congruent.

Since their corresponding parts are congruent, then we now know that side CA and side CB are congruent.

Since two sides of the triangle ABC are congruent to each other, namely side CA and side CB, then the triangle ABC is an isosceles triangle.

//

Setup for coordinate geometry proof:

M is the midpoint of AB since CM is a bisector of AB.

Since M is the midpoint of AB, then M is located at the coordinates $(\frac{0+b}{2},\frac{0+0}{2})=(\frac{b}{2},0)$ .

We found this point such that the length AM is equal to the length MB.

That is, the distance between A and M is the same as the distance between M and B.

Let's check.

$AM=\sqrt{(\frac{b}{2}-0)^2+(0-0)^2}$

$AM=\sqrt{(\frac{b}{2})^2+0}$

$AM=\sqrt{\frac{b^2}{4}}$

$AM=\frac{\sqrt{b^2}}{\sqrt{4}}$

$AM=\frac{b}{2}$

$MB=\sqrt{(b-\frac{b}{2})^2+(0-0)^2}$

$MB=\sqrt{(\frac{b}{2})^2+0}$

$MB=\sqrt{\frac{b^2}{4}}$

$MB=\frac{\sqrt{b^2}}{\sqrt{4}}$

$MB=\frac{b}{2}$

We have confirmed that AM=MB.

(Based on the picture, we could have taken a slightly easier route to calculate the distance between M and A, then the distance between B and M. They are both a horizontal distance. So $MB=b-\frac{b}{2}=\frac{b}{2}$ where as $AM=\frac{b}{2}-0=\frac{b}{2}$ .)

Now we also want to assume that the line segment CM is perpendicular to AB. I have drawn the base of the triangle on the x-axis so a vertical line would be perpendicular to it. Also this would make point $C=(c,d)=(\frac{b}{2},d)$ . The $y$ -coordinate is $d$ because we don't know how high above the $x$ -axis the point $C$ is.

If we show CA=CB, then we have shown triangle ABC is an isosceles.

Coordinate Geometry Proof:

We want to finally show that the sides CB and CA of triangle ABC are congruent. We will do this using distance formula.

That is we want to show the distance between (b/2,d) and (0,0) is the same as (b/2,d) and (b,0).

$CB=\sqrt{(b-\frac{b}{2})^2+(0-d)^2}$

$CB=\sqrt{(\frac{b}{2})^2+(-d)^2}$

$CB=\sqrt{\frac{b^2}{4}+d^2}$

$CA=\sqrt{(\frac{b}{2}-0)^2+(d-0)^2}$

$CA=\sqrt{(\frac{b}{2})^2+d^2$

$CA=\sqrt{\frac{b^2}{4}+d^2$

Thus, CA=CB. Since CA=CB, then the triangle is an isosceles.

//

3 0

4 years ago

Solve.<br> y=-x-2<br> y=4x+3

adelina 88 [10]

1) Equal the two together (not including the ‘y’)

-x -2 = 4x + 3

2) put x together and the whole number together
* when switching sides, signs ( + & -) changes, watch out for that *

-x - 4x = 3 + 2
-5x = 5 divide by the # with x so you have x
—— ——-
-5. -5
x = -1

therefore, x = -1

For y just plug x into either equation
I’ll use y = 4x + 3

y = 4(-1) + 3
y = -4 + 3
y = -1

therefore, y = -1

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3 years ago

Of the 9-letter passwords formed by rearranging the letters AAAABBCCC (4 A's, 2 B's, and 3 C's), I select one at random. Determi

Tanya [424]

Answer:

a) 3

b) (8!/9!)-(7!/9!)

c) (1-(8!/9!))*(7!/9!)

Step-by-step explanation:

a)With 4 As ; 2Bs and 3Cs it is possible to get a palindrome if you fixed the letters C according to: (2) in the extremes of the word and the other one at the center therefore you only have palindrome in the following cases

<u>C</u> ( ) <u>C</u> ( ) <u>C</u>

To fill in the gaps we have 4 letters A and 2 letters B, wich we have two divide in two palindrome gaps,

AAB and BAA the palindrome is C AAB C BAA C

BAA and AAB " " is C BAA C AAB C

ABA and ABA " " is C ABA C ABA C

b) 4 A ; 2B ; 3C

We have the total number of elements 9, so the total number of possible outcomes is : 9!

Total events: 9!

if we fixed 3 C we have (the group of 3 Cs becoming one element) so the total amount of events with 3 adjacent Cs is: 7!

Therefore the probability of having 3 adjacent Cs is: 7!/9!

If we fixed only 2 Cs we have:

4 A ; 2 B ; 2C : 1C

Total number of words (events) in this case is 8! (2C becomes 1 element)

so the total numbers of events is 8! the probability in this case is 8!/9!(this value includes cases of adjacent 3 Cs previous calculated ) so this value minus the case of 3 adjacent Cs ) give us 2 adjacent C and the other no next to them

Probability (of words with 2 adjacent Cs and the other no next to them is); 8!/9! - 7!/9!

c) Probability of B apart from each other is the whole set of events minus those where 2 B are adjacent or (become 1 element)

4 A ; 2B ; 3C

Total of events 9! and events with adjacent B is 8!/9!

Therefore the probability of words with 3 adjacent Cs and 2 B separeted is

the probability of 3 adjacent Cs (7!/9!) times probability of words with no adjacent Bs wich is (1-(8!/9!))*(7!/9!)

5 0

3 years ago

1. ABCD is a parallelogram. Find the value of x.

Deffense [45]

Answer:

Step-by-step explanation:

1. Since, it is given that ABCD is a parallelogram, therefore opposite angles of parallelogram are equal, hence

∠B=∠D

⇒7x-31=5x+13

⇒2x=44

⇒x=22

Hence, option B is correct.

2. Since, we get x=22, therefore

∠B=7x-31=7(22)-31=154-31=123°

∠B=123°

and ∠D=5x+13=5(22)+13=123°

∠D=123°

Also, ∠A+∠D=180°(corresponding angles)

⇒∠A=180-123

⇒∠A=57°

Now, ∠A=∠C=57°(opposite angles of parallelogram)

thus, Two of the angles measure 57°, and the other two measure 123°.

Therefore, option A is correct.

6 0

3 years ago

Read 2 more answers