Answer:
X = 6°
Step-by-step explanation:
In a set of parallel lines, the opposite angles and corresponding angles are similar. Since the opposite angle of 110° is the corresponding angle of 19x - 4, you can conclude that 19x - 4 = 110° ⇒ 19x = 114° ⇒ x = 6°.
Answer:
x^4 - 14x^2 - 40x - 75.
Step-by-step explanation:
As complex roots exist in conjugate pairs the other zero is -1 - 2i.
So in factor form we have the polynomial function:
(x - 5)(x + 3)(x - (-1 + 2i))(x - (-1 - 2i)
= (x - 5)(x + 3)( x + 1 - 2i)(x +1 + 2i)
The first 2 factors = x^2 - 2x - 15 and
( x + 1 - 2i)(x +1 + 2i) = x^2 + x + 2ix + x + 1 + 2i - 2ix - 2i - 4 i^2
= x^2 + 2x + 1 + 4
= x^2 + 2x + 5.
So in standard form we have:
(x^2 - 2x - 15 )(x^2 + 2x + 5)
= x^4 + 2x^3 + 5x^2 - 2x^3 - 4x^2 - 10x - 15x^2 - 30x - 75
= x^4 - 14x^2 - 40x - 75.
Answer:
![71. \ \ \ f(a) \ = \ 7a \ - \ 3; \ f(a+h) \ = \ 7a \ + \ 7h \ - \ 3; \ \displaystyle\frac{f(a+h) \ - \ f(a)}{h} \ = \ 7](https://tex.z-dn.net/?f=71.%20%5C%20%5C%20%5C%20f%28a%29%20%5C%20%20%3D%20%5C%20%207a%20%5C%20-%20%5C%203%3B%20%5C%20f%28a%2Bh%29%20%5C%20%20%3D%20%20%5C%207a%20%5C%20%2B%20%5C%207h%20%5C%20-%20%5C%203%3B%20%5C%20%5Cdisplaystyle%5Cfrac%7Bf%28a%2Bh%29%20%5C%20-%20%5C%20f%28a%29%7D%7Bh%7D%20%5C%20%3D%20%5C%207)
![72. \ \ \ f(a) \ = \ 5a^{2}; \ f(a+h) \ = \ {5a}^{2} \ + \ 10ah \ + \ {5h}^{2}; \ \displaystyle\frac{f(a+h) \ - \ f(a)}{h} \ = \ 10a \ + \ 5h](https://tex.z-dn.net/?f=72.%20%5C%20%5C%20%5C%20f%28a%29%20%5C%20%20%3D%20%5C%20%205a%5E%7B2%7D%3B%20%5C%20f%28a%2Bh%29%20%5C%20%20%3D%20%20%5C%20%7B5a%7D%5E%7B2%7D%20%5C%20%2B%20%5C%2010ah%20%5C%20%2B%20%5C%20%7B5h%7D%5E%7B2%7D%3B%20%5C%20%5Cdisplaystyle%5Cfrac%7Bf%28a%2Bh%29%20%5C%20-%20%5C%20f%28a%29%7D%7Bh%7D%20%5C%20%3D%20%5C%2010a%20%5C%20%2B%20%5C%205h)
Step-by-step explanation:
In single-variable calculus, the difference quotient is the expression
,
which its name comes from the fact that it is the quotient of the difference of the evaluated values of the function by the difference of its corresponding input values (as shown in the figure below).
This expression looks similar to the method of evaluating the slope of a line. Indeed, the difference quotient provides the slope of a secant line (in blue) that passes through two coordinate points on a curve.
.
Similarly, the difference quotient is a measure of the average rate of change of the function over an interval. When the limit of the difference quotient is taken as <em>h</em> approaches 0 gives the instantaneous rate of change (rate of change in an instant) or the derivative of the function.
Therefore,
![71. \ \ \ \ \ \displaystyle\frac{f(a \ + \ h) \ - \ f(a)}{h} \ \ = \ \ \displaystyle\frac{(7a \ + \ 7h \ - \ 3) \ - \ (7a \ - \ 3)}{h} \\ \\ \-\hspace{4.25cm} = \ \ \displaystyle\frac{7h}{h} \\ \\ \-\hspace{4.25cm} = \ \ 7](https://tex.z-dn.net/?f=71.%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cdisplaystyle%5Cfrac%7Bf%28a%20%5C%20%2B%20%5C%20h%29%20%5C%20-%20%5C%20f%28a%29%7D%7Bh%7D%20%5C%20%5C%20%3D%20%5C%20%5C%20%5Cdisplaystyle%5Cfrac%7B%287a%20%5C%20%2B%20%5C%207h%20%5C%20-%20%5C%203%29%20%5C%20-%20%5C%20%287a%20%5C%20-%20%5C%203%29%7D%7Bh%7D%20%5C%5C%20%5C%5C%20%5C-%5Chspace%7B4.25cm%7D%20%3D%20%5C%20%5C%20%5Cdisplaystyle%5Cfrac%7B7h%7D%7Bh%7D%20%5C%5C%20%5C%5C%20%5C-%5Chspace%7B4.25cm%7D%20%3D%20%5C%20%5C%207)
![72. \ \ \ \ \ \displaystyle\frac{f(a \ + \ h) \ - \ f(a)}{h} \ \ = \ \ \displaystyle\frac{{5(a \ + \ h)}^{2} \ - \ {5(a)}^{2}}{h} \\ \\ \-\hspace{4.25cm} = \ \ \displaystyle\frac{{5a}^{2} \ + \ 10ah \ + \ {5h}^{2} \ - \ {5a}^{2}}{h} \\ \\ \-\hspace{4.25cm} = \ \ \displaystyle\frac{h(10a \ + \ 5h)}{h} \\ \\ \-\hspace{4.25cm} = \ \ 10a \ + \ 5h](https://tex.z-dn.net/?f=72.%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cdisplaystyle%5Cfrac%7Bf%28a%20%5C%20%2B%20%5C%20h%29%20%5C%20-%20%5C%20f%28a%29%7D%7Bh%7D%20%5C%20%5C%20%3D%20%5C%20%5C%20%5Cdisplaystyle%5Cfrac%7B%7B5%28a%20%5C%20%2B%20%5C%20h%29%7D%5E%7B2%7D%20%5C%20-%20%5C%20%7B5%28a%29%7D%5E%7B2%7D%7D%7Bh%7D%20%5C%5C%20%5C%5C%20%5C-%5Chspace%7B4.25cm%7D%20%3D%20%5C%20%5C%20%5Cdisplaystyle%5Cfrac%7B%7B5a%7D%5E%7B2%7D%20%5C%20%2B%20%5C%2010ah%20%5C%20%2B%20%5C%20%7B5h%7D%5E%7B2%7D%20%5C%20-%20%5C%20%7B5a%7D%5E%7B2%7D%7D%7Bh%7D%20%5C%5C%20%5C%5C%20%5C-%5Chspace%7B4.25cm%7D%20%3D%20%5C%20%5C%20%5Cdisplaystyle%5Cfrac%7Bh%2810a%20%5C%20%2B%20%5C%205h%29%7D%7Bh%7D%20%5C%5C%20%5C%5C%20%5C-%5Chspace%7B4.25cm%7D%20%3D%20%5C%20%5C%2010a%20%5C%20%2B%20%5C%205h)