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____ [38]
3 years ago
13

Theme-and-variations form may be schematically outlined as ______.

Mathematics
1 answer:
lions [1.4K]3 years ago
3 0
<span>Theme-and-variations form may be schematically outlined as </span>AA'A''A'''A''''.

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Given the function f(x) = x2 and k = 3, which of the following represents the graph becoming more narrow?
Bess [88]
Functions A and C and D are translations - this means that they are exactly the same shape as f(x), but are displaced on the axes. A translation in the y-direction by k units is represented by f(x) + k, and a translation in the x-direction by k units is represented by f(x - k). This can be summarised in column vector notation:

A translation of the graph y = f(x), by the vector \left[\begin{array}{ccc}a\\b\end{array}\right] results in the graph defined by y - b = f(x - a). This is done by replacing x with (x - a) and y with (y - b). Finally it is rearranged to make y the subject, as is often the convention: y = f(x - a) + b.

Since the graph needs to change shape, this requires a 'stretch'. A stretch is performed by multiplying x or y by a constant, which here is k.

To stretch in the x-direction by a factor n, replace x with (1/n)x. Similarly to stretch in the y-direction by a factor n, replace y with (1/n)y.

For example to stretch y = f(x) by factor n in the x-direction, it would become y = f((1/n)x).

A simple quadratic graph like y = x^2 behaves slightly differently to most other graphs, in that it can narrowed by a stretch in the x or y direction. This means that a more convenient option is to perform a stretch in the y direction, since the multiplier lies outside the function notation. The stretching factor is 3, so we replace y with (1/3)y:

(1/3)y = f(x)
y = 3f(x)

Since k = 3, this becomes

y = kf(x)

... and hence the correct answer is B
6 0
3 years ago
Read 2 more answers
EXAMPLE 5 If f(x, y, z) = x sin(yz), (a) find the gradient of f and (b) find the directional derivative of f at (1, 2, 0) in the
solong [7]

Answer:

<h2>a) f =  sin(yz)i + xzcos(yz)j + xycos(yz)k</h2><h2>b) -2</h2>

Step-by-step explanation:

Given f(x, y, z) = x sin(yz), the formula for calculating the gradient of the function is expressed as ∇f(x, y, z) = fx(x, y, z)i+ fy(x, y, z)j+fz(x, y, z)k where;

fx, fy and fz are the differential of the functions with respect to x, y and z respectively.

a) ∇f(x, y, z) = sin(yz)i + xzcos(yz)j + xycos(yz)k

The gradient of f =  sin(yz)i + xzcos(yz)j + xycos(yz)k

b) Directional derivative of f at (1,2,0) in the direction of v = i + 4j − k is expressed as ∇f(1, 2, 0)*v

∇f(1, 2, 0) = sin(2(0))i +1*0cos(2*0)j + 1*2cos(2*0)k

∇f(1, 2, 0) = sin0i +0cos(0)j + 2cos(0)k

∇f(1, 2, 0) = 0i +0j + 2k

Given v = i + 4j − k

∇f(1, 2, 0)*v (note that this is the dot product of the two vectors)

∇f(1, 2, 0)*v =  (0i +0j + 2k)*(i + 4j − k )

Given i.i = j.j = k.k =1 and i.j=j.i=j.k=k.j=i.k = 0

∇f(1, 2, 0)*v = 0(i.i)+4*0(j.j)+2(-1)k.k

∇f(1, 2, 0)*v = 0(1)+0(1)-2(1)

∇f(1, 2, 0)*v =0+0-2

∇f(1, 2, 0)*v= -2

 

Hence, the directional derivative of f at (1, 2, 0) in the direction of v = i + 4j − k is -2

7 0
3 years ago
Find the area of the shaded region. Round your answer to the nearest tenth.
spayn [35]

You need to include the Image

3 0
3 years ago
A dolphin was swimming 6 feet below sea level. The number line shows the
Ksju [112]

On the number line, move 4 units to the right. End at -2. The dolphin was 2 feet below sea level.

<h3>What is number line?</h3>

A number lines are the horizontal straight lines in which the integers are placed in equal intervals.

A dolphin was swimming 6 feet below sea level.  It then swam up 4 feet.

So, -6+4= -2 feet

As, from the information

On the number line, move 4 units to the right. End at -2. The dolphin was 2 feet below sea level.

This is because the dolphin 6 feet below sea level and swam up 4 feet.

Learn more about this concept here:

brainly.com/question/14379626

#SPJ1

5 0
1 year ago
Grace sold 84 of the 140 cokies she baked what pecent did she sell
Allushta [10]
She sold 60% of her cookies.
7 0
2 years ago
Read 2 more answers
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