### 1D Solutions - Knight Math

```Name:
Date:
Period:
Assignment1D:RatesofChange
Lippman/Rasmuss
smussen
explanation,,
smuss
en textbook with as much detail, explanation
and work that is appropriate.
1.3:
1.3: 10, 12, 20, 38
Find the average rate of change of each function on the interval specified.
t 2 − 4t + 1
on [-3, 1]
t2 + 3
૚૚
The inputs ࢚ = −૜ and ࢚ = ૚ when put into the function ࢖(࢚) produce the points ((-3, ) and
10. p (t ) =
૟
૚ ૚૚
ି ି
૛ ૟
૚
(1,−
(1, ૛). The average rate of change between these two points is ૚ି(ି૜) =
૚
૝
૚૝
૜ ૚૚
ି ି
૟ ૟
૚ା૜
=
૚૝
ି
૟
=−
૝
૚૝
૟
∗
ૠ
= − ૛૝ = − ૚૛.
Find the average rate of change of each function on the interval specified. Your answers will be
expressions involving a parameter (b or h).
12. g ( x) = 2 x 2 − 9 on [4, b]
The inputs ࢞ = ૝ and ࢞ = ࢈ when put into the function ࢍ(࢞) produce the points (4,23)
(4,23) and
(࢈, ૛࢈૛ − ૢ).. ࡱ࢞࢖࢒ࢇ࢔ࢇ࢚࢏࢕࢔: ࢍ(૝) = ૛(૝)૛ − ૢ = ૛(૚૟) − ૢ = ૜૛ − ૢ = ૛૜, ࢍ(࢈) = ૛(࢈)૛ −
ૢ. The average rate of change between these two points is
૛൫࢈૛ ି૚૟൯
࢈ି૝
=
૛(࢈ା૝)(࢈ି૝)
(࢈ି૝)
൫૛࢈૛ ିૢ൯ି૛૜
࢈ି૝
=
૛࢈૛ ିૢି૛૜
࢈ି૝
=
૛࢈૛ ି૜૛
࢈ି૝
=
= ૛(࢈ + ૝).
20. g ( x) = 3x 2 − 2 on [x, x+h]
The inputs ࢞ = ࢞ and ࢞ = ࢞ + ࢎ when put into the function ࢌ(࢞) produce the points (࢞,
( ૜࢞૛ −
૛)) and (࢞ + ࢎ, ૜(࢞ + ࢎ)૛ − ૛).. The average rate of change between these two points is
൫૜(࢞ାࢎ)૛ ି૛൯ି൫૜࢞૛ ି૛൯
(࢞ାࢎ)ି࢞
૜࢞૛ ା૟ࢎ࢞ା૜ࢎ૛ ି૜࢞૛
ࢎ
=
=
൫૜(࢞ାࢎ)૛ ି૛൯ି൫૜࢞૛ ି૛൯
ࢎ
૟ࢎ࢞ା૜ࢎ૛
ࢎ
=
૜(࢞ାࢎ)૛ ି૛ ି૜࢞૛ ା૛
ࢎ
= ૟࢞ + ૜ࢎ = ૜(૛࢞ + ࢎ).
Problems from: Precalculus: An Investigation of Functions; Lippman & Rasmussen. 2014.
=
૜(࢞ାࢎ)૛ ି૜࢞૛
ࢎ
=
૜൫࢞૛ ା૛ࢎ࢞ାࢎ૛ ൯ି૜࢞૛
ࢎ
=
Use a graph to estimate the local extrema and inflection points of each function, and to estimate
the intervals on which the function is increasing, decreasing, concave up, and concave down.
38. h( x) = x 5 + 5 x 4 + 10 x 3 + 10 x 2 − 1
From the graph, we can see that the function is decreasing on the interval (−૛, ૙),, and
increasing on the intervals (−∞, −૛)‫( ܌ܖ܉‬૙, ∞).. This means that the function has a local
minimum at ࢞ = ૙, ‫ = ࢞ ܜ܉ ܕܝܕܑܠ܉ܕ ܔ܉܋ܗܔ ܉ ܌ܖ܉‬−૛. We can estimate that the function is
concave down on the interval (−∞, −૚),, and concave up on the intervals (−∞, −૚)‫( ܌ܖ܉‬૙, ∞).
This means there is an inflection point at ࢞ = −૚.
```