Answer:
The average rate of change for the given function in the interval (-5, -1) is 0 (zero)
Step-by-step explanation:
The average rate of change of a function over an interval is the quotient between the difference between the function evaluated at the ends of the interval divided by the length of the interval. That is for our case:
the average rte of change of h(t) in the interval (-5, -1) is:
so we find:
then the average rate of change becomes:
Answer:
(a) (-∞, -8) (-6, -4) (-2, ∞)
(b) -6, -2
(c) negative
(d) 5
Step-by-step explanation:
(a) A function is "decreasing" when the y-value decreases as the x-value increases.
⇒ the function is decreasing over these intervals : (-∞, -8) (-6, -4) (-2, ∞)
(b) Local maxima are the points on the function where it reaches a maximum.
⇒ maxima at x = -6 and x = -2
(c) negative
(d) 5
Answer: 3x-27
Step-by-step explanation:
(i) Velocity is the rate of change of position, so if
<em>r</em><em>(t)</em> = <em>b</em> cos(<em>ω t </em>) <em>i</em> + <em>b</em> sin(<em>ω t </em>) <em>j</em> + <em>v</em> <em>t</em> <em>k</em>
then
<em>v</em><em>(t)</em> = d<em>r</em>/d<em>t</em>
<em>v</em><em>(t)</em> = -<em>b</em> <em>ω </em>sin(<em>ω t</em> ) <em>i</em> + <em>b</em> <em>ω</em> cos(<em>ω</em> <em>t</em> ) <em>j</em> + <em>v</em> <em>k</em>
The speed of the particle is the magnitude of the velocity, given by
|| <em>v</em><em>(t)</em> || = √[(-<em>b</em> <em>ω </em>sin(<em>ω t</em> ))² + (<em>b</em> <em>ω</em> cos(<em>ω</em> <em>t</em> ))² + <em>v</em> ²]
… = √[<em>b </em>²<em>ω </em>² + <em>v</em> ²]
(ii) The path is a helix. Suppose you zero out the <em>k</em> component. Then the path is a circle of radius <em>b</em>, and the value of <em>ω</em> determines how quickly a particle on the path traverses the circle. Now if you reintroduce the <em>k</em> component, the value of <em>v</em> will determine how far from the plane <em>z</em> = 0 the particle moves in a helical path as <em>t</em> varies.
(iii) Acceleration is the rate of change of velocity, so
<em>a</em><em>(t)</em> = d<em>v</em>/d<em>t</em>
<em>a</em><em>(t)</em> = -<em>b</em> <em>ω </em>²<em> </em>cos(<em>ω t</em> ) <em>i</em> - <em>b</em> <em>ω</em> ² sin(<em>ω</em> <em>t</em> ) <em>j</em>
