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Answer:
Expression 2b+b and 3b have same values for all values of b.
Step-by-step explanation:
Given the two expression and
and values of b as 1, 2 and 3.
Consider first expression that is, .
Substituting the value of ,
....1
Substituting the value of ,
....2
Substituting the value of ,
....3
Consider second expression that is, .
Substituting the value of ,
....4
Substituting the value of ,
....5
Substituting the value of ,
....6
For value of b=1, equation 1 and equation 4 are same, for value of b=2, equation 2 and equation 5 are same and for value of b=3 equation 3 and equation 6 are same.
Therefore both expression are same for all values of b.
Answer:
see below
Step-by-step explanation:
1) see attached image
2) <u> bearings from A to B</u>
Note: bearing is always clockwise from A and from North.
(i)
bearing 180° + 67°
247° from N
(ii)
bearing 148 + 180
328° from N
(iii)
bearing 180 - 21
159° from N
(iv)
39° from N
(v)
bearing 90 - 40
50° from N
3) <u>back bearing from each diagram on no. 2</u>
<u />
Note: subtract 200 if bearing is more than 180
add 180 if the bearing is less than 180
(i) 247 - 200 = 47°
(ii) 328 - 200 = 128°
(iii) 159 + 180 = 339°
(iv) 39 + 180 = 219°
(v) 50 + 180 = 230°
Explanation:
The cubic ...
f(x) = ax³ +bx² +cx +d
has derivatives ...
f'(x) = 3ax² +2bx +c
f''(x) = 6ax +2b
<h3>a)</h3>By definition, there will be a point of inflection where the second derivative is zero (changes sign). The second derivative is a linear equation in x, so can only have one zero. Since it is given that a≠0, we are assured that the line described by f''(x) will cross the x-axis at ...
f''(x) = 0 = 6ax +2b ⇒ x = -b/(3a)
The single point of inflection is at x = -b/(3a).
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<h3>b)</h3>The cubic will have a local extreme where the first derivative is zero and the second derivative is not zero. These will only occur when the discriminant of the first derivative quadratic is positive. Their location can be found by applying the quadratic formula to the first derivative.
There will be zero or two local extremes. A local extreme cannot occur at the point of inflection, which is where the formula would tell you it is when there is only one.
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<h3>c)</h3>Part A tells you the point of inflection is at x= -b/(3a).
Part B tells you the midpoint of the local extremes is x = -b/(3a). (This is half the sum of the x-values of the extreme points.) You will notice these are the same point.
The extreme points are located symmetrically about their midpoint, so are located symmetrically about the point of inflection.
_____
Additional comment
There are other interesting features of cubics with two local extremes. The points where the horizontal tangents meet the graph, together with the point of inflection, have equally-spaced x-coordinates. The point of inflection is the midpoint, both horizontally and vertically, between the local extreme points.