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16 days ago

Consider the function f(x) = x3 − 4x2 + 2. Calculate the limit of the difference quotient at x0 = 3 for f(x).

Mathematics

1 answer:

tester [8.8K]16 days ago

3 0

Part 1.

Answer: $\boxed{f'(x)=3x^2-8x}$

The limit of the difference quotient equates to the derivative, which can be expressed as follows:

$f'(x)=\underset{\Delta x\rightarrow0}{lim}\frac{\triangle y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}$

Thus, our function is:

$f(x)=x^3-4x^2+2$

Calculating the derivative results in:

$f'(x)=3x^2-8x$

Consequently, the appropriate option is:

$\boxed{f'(x)=3x^2-8x}$

Part 2.

Answer: $\boxed{y=3x-16}$

The equation of the line that goes through the same point can be determined as:

$y-y_{0}=m(x-x_{0})$

Where $x_{0}=3$ , requiring us to find $y_{0}$ . Substitute that x-value into the function we have:

$y_{0}=f(3)=(3)^3-4(3)^2+2 \\ \\ y_{0}=-7 \\ \\ \\ So \ the \ point \ is: \\ \\ P(x_{0},y_{0})=(3,-7)$

The slope calculated is:

$m=f'(3)=3(3)^2-8(3) \\ \\ m=3$

Therefore, the equation of the line is:

$y-(-7)=3(x-3) \\ \\ \therefore y+7=3x-9 \\ \\ \boxed{y=3x-16}$

Part 3.

Answer: As illustrated below

As demonstrated below, the function's graph for $f$ is continuous. This occurs because we plotted a polynomial function whose domain encompasses all real numbers. Consequently, the function is defined at the point $P(3,-7)$ , ensuring the derivative is valid at this location, thereby allowing us to calculate the tangent line there. In summary, we present the graph below. The blue line indicates the tangent line while the red curve represents the graph of $f$ .

The appropriate expression is 1 Over 5 x Superscript minus 8 Baseline y Superscript minus 13 Baseline EndFraction

Step-by-step explanation:

In order to simplify the expression $\frac{3x^{-6}y^{-3} }{15x^{2}y^{10} }$ where x ≠ 0, y ≠ 0, we must work through the given expression.

$\frac{3x^{-6}y^{-3} }{15x^{2}y^{10} }\\= \frac{3}{15} x^{-6-2}y^{-3-10}\\ = \frac{3}{15}x^{-8}y^{-13} \\ =\frac{1}{5}x^{-8}y^{-13} \\$

The appropriate expression is 1 Over 5 x Superscript minus 8 Baseline y Superscript minus 13 Baseline EndFraction

6 0

29 days ago

A child designs a flag in the shape of a rhombus, as shown in the diagram below. Which expression can be used to determine the s

Zina [9171]

Which formula can be applied to find the side length of the rhombus?

The correct answer is the first choice: 10/Cos(30°)

Explanation:

1. The figure shows a right triangle, where the hypotenuse is denoted by "x," and this is the length you are solving for. Therefore, you have:

Cos(α)=Adjacent side/Hypotenuse

α=30°
 Adjacent side=(20 in)/2=10 in
Hypotenuse=x

2. Inputting these numbers into the equation yields:

Cos(α)=Adjacent side/Hypotenuse
 Cos(30°)=10/x

3. Hence, by isolating the hypotenuse "x," you arrive at the expression to find the side length of the rhombus, as shown below:

x=10/Cos(30°)

7 0

4 days ago

A bag of marbles can be divided evenly among 2,3 or 4 friends. A) How many marbles might be in the bag? B) What is the least num

Leona [9271]

Response:

a)12 b)12 c)36

Detailed steps:

a) Identify the least common multiple. The quantity of marbles corresponds to that value multiplied by n (with n representing an integer)

b) The least common multiple is calculated as 4*3=...

c)12*3 =?

Afterward, simply divide by 2, 3, and 4 to find the solution

6 0

7 days ago

Find the annual rate of growth (interest rate) on an account that was worth $200 in 1975 and $415.79 in 1990

tester [8842]

To find the percent change over time, use the following formula: PR = Percent Rate, VPresent = Present or Future Value, VPast = Past or Present Value. The annual percentage growth rate is calculated by dividing the percent growth by N, which is the number of years. The calculation (415.79 - 200) / 200 * 100 results in 107.89. The annual percentage growth rate is then 107.89 divided by 15, which equals 7.193.

8 0

7 days ago

Read 2 more answers

1. Darnell reads at a constant rate

Svet_ta [9500]

Answer:

Darnell reads 1,715 words in 7 minutes.

Step by step Explanation:

1. First, determine how many words he can read in a minute by dividing 735 words by 3. The result is 245.

245

______

3)735

6 drop the 3 to form 13

-_____

1 3

12 drop the 5 to make 15

-______

1 5

___________

2. Next, since he reads 735 words over 3 minutes, multiply that by 2 to find words read in 7 minutes: 3×2=6, thus, 735+735 (735×2) equals 1,470 words.

3. Finally, add 245 to account for the last minute we calculated. Therefore, the total is 1,715 words in 7 minutes.

8 0

21 day ago