Answer:
- The instantaneous rate of increase of f(x) at
is 3. - One possible equation of this line is y = 3x - 16.
- The line is tangent to the graph of y = f(x). The slope of the line is the same as the instantaneous rate of increase of f(x) at .
Step-by-step explanation:
<h3>1.</h3>
![\begin{aligned}&\lim_{h\to 0}{\frac{f(3+h)-f(h)}{h}}\\ =&\lim_{h\to 0}\frac{1}{h} \cdot [(3^{3} + 3 \times 3^{2}h +3\times 3h^{2} + h^{3})- 4(3^{2} + 2\times 3h + h^{2}) + 2 \\[-1em] &\phantom{\lim_{h\to 0}\frac{1}{h}\cdot []} -(3^{3} -4\times 3^{2} + 2)]\\ = &\lim_{h \to 0}\frac{h^{3} + 5h^{2} +3h}{h}\\=& \lim_{h \to 0}{h^{2}} + \lim_{h \to 0}{5h} + \lim_{h\to 0}{3}\\ =& 3\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%26%5Clim_%7Bh%5Cto%200%7D%7B%5Cfrac%7Bf%283%2Bh%29-f%28h%29%7D%7Bh%7D%7D%5C%5C%20%3D%26%5Clim_%7Bh%5Cto%200%7D%5Cfrac%7B1%7D%7Bh%7D%20%5Ccdot%20%5B%283%5E%7B3%7D%20%2B%203%20%5Ctimes%203%5E%7B2%7Dh%20%2B3%5Ctimes%203h%5E%7B2%7D%20%2B%20h%5E%7B3%7D%29-%204%283%5E%7B2%7D%20%2B%202%5Ctimes%203h%20%2B%20h%5E%7B2%7D%29%20%2B%202%20%5C%5C%5B-1em%5D%20%26%5Cphantom%7B%5Clim_%7Bh%5Cto%200%7D%5Cfrac%7B1%7D%7Bh%7D%5Ccdot%20%5B%5D%7D%20-%283%5E%7B3%7D%20-4%5Ctimes%203%5E%7B2%7D%20%2B%202%29%5D%5C%5C%20%3D%20%26%5Clim_%7Bh%20%5Cto%200%7D%5Cfrac%7Bh%5E%7B3%7D%20%2B%205h%5E%7B2%7D%20%2B3h%7D%7Bh%7D%5C%5C%3D%26%20%5Clim_%7Bh%20%5Cto%200%7D%7Bh%5E%7B2%7D%7D%20%2B%20%5Clim_%7Bh%20%5Cto%200%7D%7B5h%7D%20%2B%20%5Clim_%7Bh%5Cto%200%7D%7B3%7D%5C%5C%20%3D%26%203%5Cend%7Baligned%7D)
.
<h3>2.</h3>
.
In other words, the graph of y = f(x) passes through the point (3, -7) where 
.
The point-slope form of a line in a cartesian plane is:
.
For this line,
is the point on the line, while
is the slope of the line.
The equation of this line will thus be
.
That's equivalent to
.
<h3>3.</h3>
Refer to the diagram attached. The line touches the graph of y = f(x) at x = 3 without crossing it. The line here is thus a tangent to the graph of y = f(x) at x = 3. The slope of the line represents the instantaneous rate of increase of f(x) at .
Answer:
f(x) = -3x^2 + 10x + 3
Step-by-step explanation:
(0,3) (2,11)(3,6)
Using the form ax^2 + bx + c = y
From the first set we know that c = 3
Write an equation for the last two sets
x=2; y=11
a(2^2) + b(2) + 3 = 11
4a + 2b = 11 - 3
4a + 2b = 8
simplify divide by 2
2a + b = 4
and
x=3, y=6
9a + 3b + 3 = 6
9a + 3b = 6 - 3
9a + 3b = 3
simplify divide by 3
3a + b = 1
:
Use elimination to find a
3a + b = 1
2a + b = 4
--------------subtracting eliminate b
a = -3
Find b using the equation 2a + b = 4
2(-3) + b = 4
b = 4 + 6
b = 10
:
The equation: f(x) = -3x^2 + 10x + 3
Answer:
I NEED HELP! What are the dimensions of the rectangle shown below? Remember to use the axes on the coordinate grid to help you.
A coordinate grid is shown with scale from negative 14 to 0 to positive 14 on both x- and y-axes at increments of 2. A figure ABCD is shown with A at ordered pair negative 4, 4, B at ordered pair 10, 4, C at ordered pair 10, negative 4, and D at ordered pair negative 4, negative 4.
Step-by-step explanation:
Find the common ratio r for the geometric sequence, and use r to find the next 3 terms n 1, 2, 3, 4, 5, 6, 7 f(n) 1,4,16,64
mote1985 [20]
Divide the given terms:
4/1 = 4
16/4 = 4
64/16 = 4
R = 4
The next term is the previous term multiplied by 4.
64 x 4 = 256
256 x 4 = 1024
1024 x 4 = 4096
The next three terms are: 256, 1024, 4096
