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Sophie [7]
3 years ago
15

g Use this to find the equation of the tangent line to the parabola y = 2 x 2 − 7 x + 6 at the point ( 4 , 10 ) . The equation o

f this tangent line can be written in the form y = m x + b where m is: 9 Correct and where b is: Incorrect LicensePoints possible: 1 Unlimited attempts. Score on last attempt: (0.33, 0.33, 0), Score in gradebook: (0.33, 0.33, 0), Out of: (0.33, 0.33, 0.34) Message instructor about this question
Mathematics
1 answer:
natali 33 [55]3 years ago
6 0

Answer:

The tangent line to the given curve at the given point is y=9x-26.

Step-by-step explanation:

To find the slope of the tangent line we to compute the derivative of y=2x^2-7x+6 and then evaluate it for x=4.

(y=2x^2-7x+6)'          Differentiate the equation.

(y)'=(2x^2-7x+6)'       Differentiate both sides.

y'=(2x^2)'-(7x)'+(6)'    Sum/Difference rule applied: (f(x)\pmg(x))'=f'(x)\pm g'(x)

y'=2(x^2)'-7(x)'+(6)'  Constant multiple rule applied: (cf)'=c(f)'

y'2(2x)-7(1)+(6)'        Applied power rule: (x^n)'=nx^{n-1}

y'=4x-7+0               Simplifying and apply constant rule: (c)'=0

y'=4x-7                    Simplify.

Evaluate y' for x=4:

y'=4(4)-7

y'=16-7

y'=9 is the slope of the tangent line.

Point slope form of a line is:

y-y_1=m(x-x_1)

where m is the slope and (x_1,y_1) is a point on the line.

Insert 9 for m and (4,10) for (x_1,y_1):

y-10=9(x-4)

The intended form is y=mx+b which means we are going need to distribute and solve for y.

Distribute:

y-10=9x-36

Add 10 on both sides:

y=9x-26

The tangent line to the given curve at the given point is y=9x-26.

------------Formal Definition of Derivative----------------

The following limit will give us the derivative of the function f(x)=2x^2-7x+6 at x=4 (the slope of the tangent line at x=4):

\lim_{x \rightarrow 4}\frac{f(x)-f(4)}{x-4}

\lim_{x \rightarrow 4}\frac{2x^2-7x+6-10}{x-4}  We are given f(4)=10.

\lim_{x \rightarrow 4}\frac{2x^2-7x-4}{x-4}

Let's see if we can factor the top so we can cancel a pair of common factors from top and bottom to get rid of the x-4 on bottom:

2x^2-7x-4=(x-4)(2x+1)

Let's check this with FOIL:

First: x(2x)=2x^2

Outer: x(1)=x

Inner: (-4)(2x)=-8x

Last: -4(1)=-4

---------------------------------Add!

2x^2-7x-4

So the numerator and the denominator do contain a common factor.

This means we have this so far in the simplifying of the above limit:

\lim_{x \rightarrow 4}\frac{2x^2-7x-4}{x-4}

\lim_{x \rightarrow 4}\frac{(x-4)(2x+1)}{x-4}

\lim_{x \rightarrow 4}(2x+1)

Now we get to replace x with 4 since we have no division by 0 to worry about:

2(4)+1=8+1=9.

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The ratio of males to females in a cycling club is 5:3.
taurus [48]

Answer:

3/2

Step-by-step explanation:

The ratio of males to females in a cycling club is 5:3.

1/3 of the males are under 18

2/9 of the females are under 18

The fraction of the club members that are under 18 is calculated as:

Male: Female = Male/Female

We are told in the above question:

1/3 of the males are under 18

2/9 of the females are under 18

Hence:

1/3 / 2/9

= 1/3 ÷ 2/9

= 1/3 × 9/2

= 3/2

The fraction of the club members that are under 18 is 3/2

6 0
3 years ago
Help 8th grade math need answers quick
stepladder [879]

Answer:

A few examples:

0. 0

1. Undefined

2. -3/5

3. Undefined

...

7. 493

8. 5/2

....

Learn how to find the slope below for the other problems.

Step-by-step explanation:

Slope is the rate of change for a linear function. It is found by subtracting the y values of two point on the line and dividing that difference by the difference of the x values of the points.

It can also be found using the formula y=mx+b known as the slope intercept form.

Here are a few examples:

0. This is a horizontal line which always has slope 0.

1. This is a vertical line which always has slope undefined.

2. Find two points that cross through a grid line intersection The line appears to cross them at (5,3) and (0,6). Count the unit squares between the two by counting up 3 and over to the left 5. Because it is left it is negative. The slope is -3/5

3. To find the slope, use the slope formula:

\frac{y_2-y_1}{x_2-x_1}=\frac{6--4}{-2--2}=\frac{6+4}{-2+2}=\frac{10}{0}=undefined

Since we can't divide by 0, it is undefined.

7. y=493x-257 follows the formula y=mx+b where m is the slope. m=493. The slope is 493.

8. Covert the equation into y=mx+b by rearranging the terms using y=mx+b.

5x-2y=48

-2y=48-5x

y=5/2 x -24

So the slope is 5/2.

8 0
3 years ago
Find the slope between the pair of points (1,3) and (7,2)
agasfer [191]

Answer:

             \bold{m=-\dfrac16}

Step-by-step explanation:

\bold{slope\, (m)=\dfrac{change\ in\ Y}{change\ in\ X}=\dfrac{y_2-y_1}{x_2-x_1}}

(1, 3)    ⇒   x₁ = 1,  y₁ = 3

(7, 2)    ⇒   x₂ = 7,  y₂ = 2

So the slope:

                    \bold{m=\dfrac{2-3}{7-1}=\dfrac{-1}6=-\dfrac16}

5 0
3 years ago
What is the domain of the function graphed below?
I am Lyosha [343]

Answer:

x < 7

Step-by-step explanation:

the domain is the number range or interval of x the function is defined for to generate a y-value.

the graph shows that the y-values continue with x-values to the right up to but not including 7 (the empty dot says "excluding"). and with x- values to the left to negative infinity.

x=-1 is not a problem, because while for the left part of the function it is not included, but for the right part it is.

so, it is fully defined for all x < 7.

4 0
3 years ago
Find the roots x^2-13x+30=0<br> (numbers only)
Morgarella [4.7K]

Answer:

Find the roots of x^2-13x+30=0 by solving for x

x= 10,3

Step-by-step explanation:

3 0
3 years ago
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