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max2010maxim [7]
3 years ago
11

Graph the line. y - 2 = -3(x + 3)

Mathematics
1 answer:
Tatiana [17]3 years ago
3 0

Answer:

  see below

Step-by-step explanation:

The equation is in point-slope form, so you know the line goes through the point (x, y) = (-3, 2) and it has a slope of -3. The slope tells you the line goes down 3 units for each 2 units to the right.

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The difference of 3/8 of 1/2 and 3/4 of 1/3
Rainbow [258]

Answer:

difference between 3/8 and 1/2 is 1/8

difference between 3/4 and 1/3 is

5/12 i believe

Step-by-step explanation:

just change to common denominator and find the difference

5 0
3 years ago
HELP MEeeeeeeeee g: R² → R a differentiable function at (0, 0), with g (x, y) = 0 only at the point (x, y) = (0, 0). Consider<im
GrogVix [38]

(a) This follows from the definition for the partial derivative, with the help of some limit properties and a well-known limit.

• Recall that for f:\mathbb R^2\to\mathbb R, we have the partial derivative with respect to x defined as

\displaystyle \frac{\partial f}{\partial x} = \lim_{h\to0}\frac{f(x+h,y) - f(x,y)}h

The derivative at (0, 0) is then

\displaystyle \frac{\partial f}{\partial x}(0,0) = \lim_{h\to0}\frac{f(0+h,0) - f(0,0)}h

• By definition of f, f(0,0)=0, so

\displaystyle \frac{\partial f}{\partial x}(0,0) = \lim_{h\to0}\frac{f(h,0)}h = \lim_{h\to0}\frac{\tan^2(g(h,0))}{h\cdot g(h,0)}

• Expanding the tangent in terms of sine and cosine gives

\displaystyle \frac{\partial f}{\partial x}(0,0) = \lim_{h\to0}\frac{\sin^2(g(h,0))}{h\cdot g(h,0) \cdot \cos^2(g(h,0))}

• Introduce a factor of g(h,0) in the numerator, then distribute the limit over the resulting product as

\displaystyle \frac{\partial f}{\partial x}(0,0) = \lim_{h\to0}\frac{\sin^2(g(h,0))}{g(h,0)^2} \cdot \lim_{h\to0}\frac1{\cos^2(g(h,0))} \cdot \lim_{h\to0}\frac{g(h,0)}h

• The first limit is 1; recall that for a\neq0, we have

\displaystyle\lim_{x\to0}\frac{\sin(ax)}{ax}=1

The second limit is also 1, which should be obvious.

• In the remaining limit, we end up with

\displaystyle \frac{\partial f}{\partial x}(0,0) = \lim_{h\to0}\frac{g(h,0)}h = \lim_{h\to0}\frac{g(h,0)-g(0,0)}h

and this is exactly the partial derivative of g with respect to x.

\displaystyle \frac{\partial f}{\partial x}(0,0) = \lim_{h\to0}\frac{g(h,0)-g(0,0)}h = \frac{\partial g}{\partial x}(0,0)

For the same reasons shown above,

\displaystyle \frac{\partial f}{\partial y}(0,0) = \frac{\partial g}{\partial y}(0,0)

(b) To show that f is differentiable at (0, 0), we first need to show that f is continuous.

• By definition of continuity, we need to show that

\left|f(x,y)-f(0,0)\right|

is very small, and that as we move the point (x,y) closer to the origin, f(x,y) converges to f(0,0).

We have

\left|f(x,y)-f(0,0)\right| = \left|\dfrac{\tan^2(g(x,y))}{g(x,y)}\right| \\\\ = \left|\dfrac{\sin^2(g(x,y))}{g(x,y)^2}\cdot\dfrac{g(x,y)}{\cos^2(g(x,y))}\right| \\\\ = \left|\dfrac{\sin(g(x,y))}{g(x,y)}\right|^2 \cdot \dfrac{|g(x,y)|}{\cos^2(x,y)}

The first expression in the product is bounded above by 1, since |\sin(x)|\le|x| for all x. Then as (x,y) approaches the origin,

\displaystyle\lim_{(x,y)\to(0,0)}\frac{|g(x,y)|}{\cos^2(x,y)} = 0

So, f is continuous at the origin.

• Now that we have continuity established, we need to show that the derivative exists at (0, 0), which amounts to showing that the rate at which f(x,y) changes as we move the point (x,y) closer to the origin, given by

\left|\dfrac{f(x,y)-f(0,0)}{\sqrt{x^2+y^2}}\right|,

approaches 0.

Just like before,

\left|\dfrac{\tan^2(g(x,y))}{g(x,y)\sqrt{x^2+y^2}}\right| = \left|\dfrac{\sin^2(g(x,y))}{g(x,y)}\right|^2 \cdot \left|\dfrac{g(x,y)}{\cos^2(g(x,y))\sqrt{x^2+y^2}}\right| \\\\ \le \dfrac{|g(x,y)|}{\cos^2(g(x,y))\sqrt{x^2+y^2}}

and this converges to g(0,0)=0, since differentiability of g means

\displaystyle \lim_{(x,y)\to(0,0)}\frac{g(x,y)-g(0,0)}{\sqrt{x^2+y^2}}=0

So, f is differentiable at (0, 0).

3 0
3 years ago
If f(x) = x2 + x, find f(-3).<br><br> A. -12<br> B. 3<br> C. 6
Alex17521 [72]

C.


-3 squared is 9. 9 + (-3) is 6.


Therefore, it's C

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scZoUnD [109]

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