Hello, I need some assistance with this homework question please for precalculusHW 23
Answers
ANSWER
slope = 1
EXPLANATION
Given:
Points (1, 2) and (8, 9).
Desired Outcome:
Slope of the line
Applying the slope formula
[tex]slope\text{ = }\frac{y_2\text{ - y}_1}{x_2\text{ - x}_1}[/tex]where:
y2 = 9,
y1 = 2
x2 = 8 and
x1 = 1
Substituting the values
[tex]\begin{gathered} slope\text{ = }\frac{9\text{ - 2}}{8\text{ - 1}} \\ slope\text{ = }\frac{7}{7} \\ slope\text{ = 1} \end{gathered}[/tex]Hence, the slope of the line containing the points (1, 2) and (8, 9) is 1.
Related Questions
Find the area of the figure. Use 3.14 for .18 in9 inO A. 97.2 in2O B. 122.24 in2O C. 61.12 in2D. 86.24 in2
Answers
SOLUTION
We want to solve the question below
The figure consists of a semi-circle and a triangle. So the area of the figure becomes
Area of semi-circle + area of triangle
The semi-circle has a diameter of 8 in. So the radius becomes
[tex]r=\frac{diameter}{2}=\frac{8}{2}=4in[/tex]Area of the semi-circle is given as
[tex]\begin{gathered} \frac{1}{2}\times\pi r^2 \\ \frac{1}{2}\times3.14\times4^2 \\ \frac{1}{2}\times3.14\times16 \\ =25.12\text{ in}^2 \end{gathered}[/tex]Area of the triangle is
[tex]\begin{gathered} \frac{1}{2}\times base\times height \\ \frac{1}{2}\times9\times8 \\ 9\times4 \\ =36\text{ in}^2 \end{gathered}[/tex]So Area of the figure becomes
[tex]\begin{gathered} 25.12+36 \\ =61.12\text{ in}^2 \end{gathered}[/tex]Hence the answer is option C
at the time of the weather forecast on Evening News, the temperature was 4 degrees below zero. The temperature continue to fall at a rate of 5 degrees each hour or due to a winter storm. Which equation represents the relationship between the temperature t, in degrees after h hours
Answers
Reasoning Krishan wants his quiz average to be at least90 so that he can get an A in the class. His current quiz scoresare: 80, 100, 85. What does he have to get on his 2052next quiz to have an average of 90?A 85B 90C 92D 95
Answers
Average = total of all test scores/number of tests
90 = ( 80 + 100 + 85 + x ) / 4
Solve for x
90 (4) = 80 + 100 +85 + x
360 = 265 + x
360-265 = x
95 = x
Circle describe and correct each error Graph y=x-4 using slop-intercept form.M= -4y-int=1
Answers
Given:
Given that a graph of the function
[tex]\begin{gathered} y=x-4 \\ m=-4 \\ y-int=1 \end{gathered}[/tex]Required:
To find error in the given question.
Explanation:
The standard equation of the line is
[tex]y=mx+c[/tex]Where m slope and c is y-intercept.
Consider the given equation
[tex]y=x-4[/tex]Here the slope is 1 and y-intercept is at -4.
And the graph of the equation is,
Final Answer:
The error is :
[tex]\begin{gathered} m=-4 \\ y-int=1 \end{gathered}[/tex]Where in the xy-plane are the points with x < 0 and y is greater than or equal to 0?*O Quadrant IO Quadrant IIO Quadrant IIIO Quadrant IV
Answers
Answer:
Quadrant II
Explanation:
In the xy-plane:
• The value of x is less than 0 in Quadrant II and Quadrant III.
,• The value of y is greater than or equal to 0 in Quadrant I and Quadrant II.
Therefore, the quadrant with points x < 0 and y≥0 is Quadrant II.
which of the following statements correctly compares the tow functions f(x) and g(x)?.
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We have two functions and we have to find which statements are true.
They both have a maximum value of 1.
f(x) has a minimum and not a maximum, so this statement is not true.
The graphs of both functions cross the x-axis at 0.
f(x) does not cross the x-axis, so this statement is not true.
The graphs of both functions cross the y-axis at 1.
This is true for f(x).
For g(x), we have to calculate g(0) to find at which value of y the function cross the y-axis:
[tex]g(0)=-4\cdot0^2+1=0+1=1[/tex]This statement is true.
Function f(x) has a minimum value of 1 and function g(x) has a maximum value of 1.
This is true for f(x).
For g(x), the maximum value happens when x=0, because for all other values of x, the quadratic term becomes more negative.
In the previous statement we calculate g(0)=1, so 1 is the maximum value of g(x).
This statement is true.
They both have a minimum value of 1.
g(x) does not have a minimum value. This statement is not true.
Answer: The statement that are true:
- The graphs of both functions cross the y-axis at 1.
- Function f(x) has a minimum value of 1 and function g(x) has a maximum value of 1.
Solve Each System by Elimination:-3x-5y=14-5x+7y=8
Answers
(-3, -1)
1) Solving this system by Elimination method:
Let's eliminate the x variables firstly:
-3x-5y=14 x -5 Multiply by the factor that yields the LCM (3,5) =15
-5x+7y=8 x 3 Multiply by the factor that yields the LCM (3,5) =15
15x +25y =-70
-15x +21y =24 Add both equations simultaneously
-----------------------
46y= -46 Divide both sides by 46
y= -1
2) Plug y=-1, into the smaller coefficients equation, just for convenience
-3x -5y = 14
-3x -5(-1) = 14
-3x +5=14 Subtract 5 from both sides
-3x = 9
x= -3
3) So the answer to this Linear System is (-3, -1)
a man filled his car's 16 galllon gas tank. he took a trip and used 1/2 of the gas. how many gallons of gas were used?
Answers
Given:
The capacity of the gas tank = 16 gallon
He filled the gas tank and used half of it for a trip i.e
fraction of gallon used = 1/2
Solution
The gallon of gas used can be calculated using the formula:
[tex]\text{gallon of gas used = fraction of gallon used }\times\text{ gallon of gas filled}[/tex]Substituting, we have:
[tex]\begin{gathered} \text{gallon of gas used = }\frac{1}{2}\text{ }\times\text{ 16} \\ =\text{ 8 gallons} \end{gathered}[/tex]Answer: 8 gallo
A farmer has 1776 feet of fencing available to enclose a rectangular area bordering a river. If no fencing is required along the river, find the dimensions of the fenced area that will maximize the area. What is the maximum area?
Answers
As per the given perimeter of the rectangular area, the maximum area without fencing is 788544 square feet.
Perimeter of rectangle
Perimeter of the rectangle is defined as the total length or distance around the boundary of a rectangle.
And the formula that is used to measure the perimeter of the rectangle is
P = L x B
Where
L refers the length
B refers the breadth
Given,
A farmer has 1776 feet of fencing available to enclose a rectangular area bordering a river.
Here we need to find if no fencing is required along the river, then what will be the dimensions of the fenced area that will maximize the area.
Let us consider L and W be the length and width of the rectangular respectively.
And also, let the river run along L.
So, the perimeter to be covered by fence is written as,
=> P = L + 2W.
Therefore, when we apply the value of perimeter in it, then we get,
=> 1776 =L + 2W
Here we need the value of L, so, the equation is rewritten as,
=> L = 1776 - 2W
Now, we have to apply these value on the area formula, then we get,
A = (1776-2W) x W
When we simplify it, then we get,
=> A = 2700W-2W²
This is in the form of quadratic equation.
So, let us assume that the vertex of the rectangular at maximum area will give maximum width.
Then it can be obtained as, (W,A),
where the value of
W = -b/2a
Here the value of b = 1776 and a = -2
By applying these values on the formula, then we get the value of W as,
=> W = -1776/2*(-2)
=> W = -1776/-4
=> W = 444ft.
Therefore, the length is
=> L = 1776 - 2(444)
=> L = 1776 - 888
=> L = 888
Maximum area, A=888*888 = 788544 square feet.
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Help on any of these problems would be appreciated. Thanks! Question 1
Answers
Theorem: The measure of the angle at the center is equal to the measure of the angle at the circumference.
Hence, the answer is
[tex]x=70^0[/tex]Y is inversely proportional to the cube of x. If Y = 5 when x = 2, then k = 20.
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Answer:
False
Explanation:
Y is inversely proportional to the cube of x. Mathematically, this means
[tex]y\propto\frac{1}{x^3}[/tex]If we now introduce a proportionality constant k, then we get
[tex]y=\frac{k}{x^3}[/tex]Now if y = 5 when x = 2, then
[tex]5=\frac{k}{2^3}[/tex][tex]5=\frac{k}{8}[/tex]Multiplying both sides by 8 gives
[tex]5\times8=k[/tex][tex]\boxed{k=40.}[/tex]Hence, the value of k is NOT 20.
Therefore, the statement that "If Y = 5 when x = 2, then k = 20." is false.
4. I The coordinates of ΔLMN are L(0,-3), M(2,1) and N(7,0). Right the coordinates of L’,M’, and N’ when ΔLMN is under a translation 2 units to the left and 4 units up
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Given: The coordinate of triangle LMN as
[tex]\begin{gathered} L(0,-3) \\ M(2,1) \\ N(7,0) \end{gathered}[/tex]To Determine: The coordinates of the image, L'M'N' under the translation 2 units to the left and 4 units up
The translation rule for a translation of of a units to the left is
[tex](x,y)\rightarrow(x+a,y)[/tex]The translation rule for translation of b units up is
[tex](x,y)\rightarrow(x,y+b)[/tex]Therefore, the translation rule of a units to the left and b units up is
[tex](x,y)\rightarrow(x+a,y+b)[/tex]Applying the rule to given translation of 2 units to the left and 4 units up would be
[tex](x,y)\rightarrow(x+2,y+4)[/tex]Now, we apply the rule to get the coordinates of the image as shown below
[tex]\begin{gathered} L(0,-3)\rightarrow L^{\prime}(0+2,-3+4)=L^{\prime}(2,1) \\ M(2,1)\rightarrow M^{\prime}(2+2,1+4)=M^{\prime}(4,5) \\ N(7,0)\rightarrow N^{\prime}(7+2,0+4)=N^{\prime}(9,4) \end{gathered}[/tex]Hence, the coordinate of the image is
L'(2,1)
M' (4,5)
N' (9,4)
Problem 2: Find mZH. H 89° 5.r-7 Exterior angle: D and Remote angles: Equation:
Answers
The exterior angle is
Remote angles are ; 70° and 50°
Exterior angle is equal to the sum of the opposite interior angles
Hence;
Question 9 Which equation would generate the arithmetic sequence: -5,- 14,- 23,-32,-41, A an = -5+9(n-1) B an = -5-9(n-1) C None of the other answers are correct D an = 9+5(n-1) E a = 9-5(n-1)
Answers
B
1) Examining the Arithmetic Sequence:
(-5,-14, -23,-32,-41,..)
We have the following information:
a_1 = -5
Common ratio:-9
2) From these data, we have the Explicit formula:
[tex]a_n=-5_{}-9(n-1)[/tex]3) Looking at the options we have some formulas, so we can state that the equation that would generate the Arithmetic Sequence is described in option B
Helpppppp plas I don’t know the answer and I’m crying
Answers
The given expressions are:
A: 3(x+2)
B: 3x+6
When two expressions are equivalent, it doesn't matter which x-value you use, the result will always be the same, then option C and option D show that these expressions are equivalent, because we will obtain the same result, regardless of the value of x.
Now, the distributive property states:
a(x+y)=a*x+a*y
If we apply this property to expression A, we have:
3*x+3*2=3x+6
Thus, by applying the distributive property we can see that those expressions are equivalent.
The option B which says "Both expressions involve addition" does not show that these expressions are equivalent since we can have different expressions as: 4x+8, x+1, etc... and these are not equivalent.
Thus, the statement which doesn't show that these expressions are equivalent is B. Both expression involve addition..
Now, the
Consider the linear equation 2y - 3x = 5.Are (-1, 1) and (4, 1) solutions to the inequality 2y - 3x < 5? Explain how you know.
Answers
Solution
For this case we have the following inequality:
2y-3x< 5
And we can solve for y like this:
2y < 3x+5
y < 1/2 (3x+5)
We can replace the points and we can verify:
x=-1 y=1/2*(3*-1 +5) = 1/2(-3+5)= 1 then y is not <1
x=4 y=1/2*(3*4 +5) = 1/2(12+5)= 17/2 then y is not <1
Lori has purple and red flowers in groups of 6. She has x groups of purple flowers and y groups of red flowers. Select an expression that shows the total number of flowers that Lori has?
Answers
Given
the number of group of purple as x
the number of group of red as y
The sum total of the flowers in group of 1 will be x+y
Since we are to find the total number of flowers she will have in group of 6, we will multiply the sum of the flower by 6 as shown;
6(x+y)
Open the parenthesis
= 6(x)+6(y)
= 6x+6y
The correct option is B
What is the exact value of cosine of the quantity pi over 3 question mark
Answers
We are required to find the value of the cosine of pi over 3.
The cosine of an angle is a ratio of the side adjacent to the angle to the hypothenuse side
Our approach is to first plot a triangle that will help us give values to this side and get our ratio.
Fortunately, pi over 3 is a special angle as we will see.
We can convert it to degrees via the formula:
[tex]\frac{\pi}{3}\times\frac{180^o}{\pi}=60^o[/tex]Recall that the sum of angles in an equilateral triangle of sides' ratio 2:2:2 is 180 degrees and each angle is 60 degrees.
We can find side o through Pythagoras Theorem as:
[tex]o=\sqrt[]{2^2-1^2}=\sqrt[]{4-1}=\sqrt[]{3}[/tex]The cosine of the angle is a ratio of the adjacent side, o and hypothenuse, 2.
[tex]\cos 60^o=\cos \frac{\pi}{3}=\frac{\text{adj}}{\text{hyp}}=\frac{1}{2}[/tex]OPTION B
The diagonals of a parallelogram are 56 in and 34 in and intersect at angle of 120° find the length of the shorter side
Answers
Diagonals of a parallelogram bisect each other.
The opposite sides of a parallelogram are parallel and equal.
In a triangle, the larger angle has a longer opposite side and a smaller angle has a shorter opposite side.
Law of cosine: If a, b, c are three sides of a triangle and A is the angle opposite to the side a, then
[tex]a^2=b^2+c^2-2bc\cos A[/tex]The diagonals of a parallelogram are 56 inches and 34 inches. They bisect each other and form 4 triangles.
Let ABCD is a parallelogram and the diagonals AC and BD intersect each other at point O.
AB parallel to CD , AB=CD.
BC parallel to AD , BC=AD.
Diagonals intersect at an angle of 130 degrees.
m∠AOD=120 degree.
BD is a straight line. So,
m∠AOD+m∠AOB=180 degree
120+m∠AOB=180 degree
∠AOB =180-120=60 degree.
The opposite side of 130∘, (AD and BC) are the longer sides and the opposite side of 60∘, (AB and CD) are the shorter sides.
Use the law of cosine in triangle AOB,
[tex]AB^2=OA^2+OB^2+2(OA)(OB)\cos 60^{\circ}[/tex][tex]AB^2=28^2+17^2+2\times28\times17\cos 60^{\circ}[/tex][tex]AB^2=784+289+476[/tex][tex]AB^2=1549[/tex][tex]AB=39.35\text{ in}[/tex]The length of shorter side is AB =39.35 in.
Noah has a coupon for 30% off at his favorite clothing store can you use it to buy a hoodie and a pair of jeans I paid $28 for the jeans after using the coupon what is the regular price
Answers
Given:
Coupon = 30%
Amount paid after using the coupon = $28
Let's find the regular price.
The coupon is a form of voucher that enables someone to get a discount off a product.
This means after a discount of 30%, the new price of the jeans is $28
Thus, to find the regular price, we have:
[tex]28=P(1-\frac{30}{100})[/tex]Where P represents the regular price.
From the equation above, let's solve for P.
[tex]28=P(1-0.3)[/tex][tex]\begin{gathered} 28=P(0.7) \\ \\ 28=0.7P \\ \\ \text{Divide both sides by 0.7:} \\ \frac{28}{0.7}=\frac{0.7P}{0.7} \\ \\ 40=P \\ \\ P=40 \end{gathered}[/tex]Therefore, the regular price for the Jeans is $40
ANSWER:
$40
Jessica is a professional baker. She bakes 113 cupcakes in 2 hours How many cupcakes will she make in 6 hours? Jessica can make cupcakes in 6 hours How long will it take her to make 791 cupcakes? It will take Jessica 791 cupcakes hours to make The equation that represents this situation is y Time (hour) Cupcakes 113 2 791
Answers
You know that Jessica can bake 113 cupcakes in 2hours, using this relationship you can calculate the number of cupcakes she can bake in 6 hours using cross multiplication:
2hours_____113cupcakes
6hours_____xcupcackes
[tex]\begin{gathered} \frac{113}{2}=\frac{x}{6} \\ (\frac{113}{2})\cdot6=x \\ 339=x \end{gathered}[/tex]She can bake 339 cupcakes in 6 hours.
*-*-*-*-*-*-*
To determine how much time it will take to make 791 cupcakes you can also apply cross multiplication, this time you know the amounts of cupcakes and neet to calculate the time:
So if the can make 113 cupcakes in 2 hours,
Then she will make 791 cupcakes in x hours:
113 cupcakes_____2hours
791 cupcakes_____xhours
[tex]\begin{gathered} \frac{2}{113}=\frac{x}{791} \\ (\frac{2}{113})\cdot791=x \\ 14=x \end{gathered}[/tex]It will take her 14 hours to make 791 cupcakes.
*-*-*-*-*-*-*
To determine an equation that represents this situation, first determine the variables.
In this case:
y → will represent the number of cupcakes made
x → will represent the time she spent coocking the cupcakes
Next is to determine how many cupcakes she makes in one hour:
If she makes 113 cupcakes in 2hours, in half the time she will make half the cupcakes, that is
[tex]\frac{113}{2}=56.5[/tex]She makes 56.5 cupcakes per hour, since each passing hour se adds 56.5 cupcakes then this number will represent the coefficient of variation (or slope) of the equation and must multiply x.
Then the equation that represents this relationship is
[tex]y=56.5x[/tex]Find the area between the graph of y= -12x^3 and the x-axis on the interval [-1, 1]. Write the exact answer. Do not round.
Answers
Recall that the integral of the area between the graph of two functions, in an interval [a,b] is:
[tex]\int ^b_a|f(x)-h(x)|dx\text{.}[/tex]Now, if f(x) is an odd function, we can use the following property:
[tex]\int ^a_{-a}|f(x)|dx=2\int ^a_0|f(x)|dx\text{.}[/tex]Now, notice that the function y=-12x³ is an odd function, therefore:
[tex]\int ^1_{-1}|y-0|dx=2\int ^1_0|-12x^3|dx=2\int ^1_012x^3dx\text{.}[/tex]Applying the linearity of the integral we get:
[tex]24\int ^1_0x^3dx=24\frac{x^4}{4}|^1_0=24(\frac{1}{4}-0)=\frac{24}{4}=6.[/tex]Answer: 6.
answer in standard form and contain only positive(x+2) (2x^2-x-9)
Answers
(x+2) (2x^2-x-9)
Apply distributive property:
x(2x^2)+x (-x) + x (-9) + 2 (2x^2) + 2 (-x) + 2 (-9)
2x^3 - x^2 - 9x + 4x^2 - 2x - 18
Combine like terms:
2x^3 -x ^2 + 4x^2 - 9x -2x -18
2x^3 + 3x^2 - 11x - 18
Write an equation in standard form of the line passing through the points (12, 6) and (-4, 10).The equation is . (Type your answer in standard form.)
Answers
Answer:
The equation, in the standard form is: x + 4y = 36
Step-by-step explanation:
The standard form of the equation of a line has the following format:
Ax + By = C.
First, I will place the equation of the line in slope-intercept formula, which is:
y = ax + b. Then, I pass to the standard.
Passes through the point (12,6):
This means that when x = 12, y = 6.
So
y = ax + b
6 = 12a + b
b = 6 - 12a
Passes through the point (-4,10):
This means that when x = -4, y = 10. SO
10 = -4a + b
Since b = 6 - 12a
10 = -4a + (6 - 12a)
10 = -4a + 6 - 12a
10 - 6 = -4a - 12a
-16a = 4
16a = -4
a = -4/16
Simplifying by 4
a = -1/4
b = 6 - 12a = 6 - 12*(-1/4) = 6 + 3 = 9
So
y = ax + b
y = -(x/4) + 9
(x/4) + y = 9
Multiplying everything by 4
x + 4y = 36
The equation, in the standard form is: x + 4y = 36
11. Using the diagram below, classify the angle pairs as corresponding. alternate interior, alternate exterior, consecutive interior, consecutive exterior, or none.a. < 6 and < 7
Answers
• 1 and 3.
,• 9 and 11.
,• 2 and 4.
,• 10 and 12.
,• 5 and 7.
,• 13 and 15.
,• 6 and 8.
,• 14 and 16.
,• 1 and 5.
,• 2 and 6.
,• 9 and 13.
,• 10 and 14.
,• 3 and 7.
,• 4 and 8.
,• 11 and 15.
,• 12 and 16.
The alternate interior angles are• 9 and 4.
,• 10 and 3.
,• 13 and 8.
,• 14 and 7.
,• 2 and 13.
,• 10 and 5.
,• 4 and 15.
,• 12 and 7.
The alternate exterior angles are• 1 and 12.
,• 2 and 11.
,• 5 and 16.
,• 6 and 15.
,• 1 and 14.
,• 9 and 6.
,• 3 and 16.
,• 11 and 8.
The consecutive interior angles are• 9 and 3.
,• 10 and 4.
,• 13 and 7.
,• 14 and 8.
,• 2 and 5.
,• 10 and 13.
,• 4 and 7.
,• 12 and 15.
The consecutive exterior angles are• 1 and 11.
,• 2 and 12.
,• 5 and 15.
,• 6 and 16.
,• 1 and 6.
,• 11 and 16.
,• 9 and 14.
,• 11 and 16.
Therefore, angles 6 and 7 are none of the choices.By the congruent supplements theorem, what can youconclude?CBG = _DBGO_FBC = _DBG_CBG is supplementary to _DBF._FBC is supplementary to DBG.
Answers
The congruent supplements theorem basically states that if we have two pairs of supplementary angles, say A and B are supplementary and C and D are supplementary and one of angle of each pair are congruent, say A congruent to C, then the other two are also congruent (say B congruent to D)
In our case, the angle A is the angle FBC, B is the angle CBG. C is the angle DBG and D is the angle DBF. Since B is congruent to D then A is congruent to C. So angle FBC is congruent to angle DBG, which is option 2
Suppose that y varies directly with x and y = 2 when x =16 write a direct variation equation that relates x and y
Answers
The equation of the direct variation is expressed as: y = 1/8x.
How to Write a Direct Variation Equation?The equation of a direct variation between two variables, say x and y, is expressed as y = kx, where k is the constant of proportionality, if y varies directly as x.
Therefore, substitute y = 2 and x = 16 into y = kx to find the value of k:
2 = k(16)
Divide both sides by 16
2/16 = k
1/8 = k
k = 1/8.
To write the equation of the direct variation, substitute k = 1/8 into y = kx:
y = 1/8x.
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How many degrees would this octagon need to be rotated clockwise around its center to get point K to point G
Answers
In the image below you can observe that we have to rotate four times.
Where each rotation is 45 degrees. So,
[tex]45\times4=180[/tex]Hence, the right answer is A. 180°.The half-life is blank years. Round to one decimal place as needed
Answers
SOLUTION
The formula to apply is
[tex]\begin{gathered} A=A_oe^{-\lambda t} \\ Where\text{ A = amount of substance remaining = 0.5, after half decayed} \\ A_o=1 \\ \lambda=0.051 \\ t=\text{ time in years } \end{gathered}[/tex]Putting in the values into the formula, we have
[tex]\begin{gathered} 0.5=1\times e^{-0.051t} \\ 0.5=e^{-0.051t} \\ Taking\text{ ln of both sides, we have} \\ ln0.5=-0.05t \\ t=\frac{ln0.5}{-0.051} \\ t=13.59112 \end{gathered}[/tex]Hence the answer is 13.6 years to 1 d.p
f(x) = 6x^4 + 6Use the limit process to find the slope of the line tangent to the graph of f at x = 2. Slope at x= 2:__Find an equation of the line tangent to the graph of f at x = 2:__
Answers
The given function is
f(x) = 6x^4 + 6
The formula for the limit is shown below
[tex]\begin{gathered} f^{\prime}(x)\text{ = }\lim _{h\to0}\text{ }\frac{f(x\text{ + h) - f(x)}}{h} \\ \text{Substituting x = x + h into the function, we have} \\ f^{\prime}(x)\text{ = }\lim _{h\to0}\text{ }\frac{6(x+h)^4+6-(6x^4+6)}{h} \\ f^{\prime}(x)\text{ = }\lim _{h\to0}\text{ }\frac{6(h^4+4h^3x+6h^2x^2+4hx^3+x^4)+6-6x^4-6}{h} \\ f^{\prime}(x)\text{ = }\lim _{h\to0}\text{ }\frac{6h^4+24h^3x+36h^2x^2+24hx^3+6x^{4\text{ }}-6x^4\text{ + 6 - 6}}{h} \\ f^{\prime}(x)\text{ = }\lim _{h\to0}\text{ }\frac{h(6h^3+24h^2x+36hx^2+24x^3)}{h} \\ h\text{ cancels out} \\ \end{gathered}[/tex]Evaluating the limit at h = 0, we would substitute h = 0 into 6h^3 + 24h^2x + 36hx^2 + 24x^3
It becomes
6(0)^3 + 24(0)^2x + 36(0)x^2 + 24x^3
The derivative is 24x^3
f'(x) = 24x^3
This is the slope of the tangent line is at x = 2
By substituting x = 2 into f'(x) = 24x^3, it becomes
f'(2) = 24(2)^3 = 192
To find the y coordinate of the point, we would substitute x = 2 into
f(x) = 6x^4 + 6
y = 6(2)^4 + 6 = 102
Thus, the x and y coordinates are (2, 102) and the slope is 192
The equation of the line in the point slope form is
y - y1 = m(x - x1)
Thus, the equation of the tangent is
y - 102 = 192(x - 2)