In Exercises 7 − 12 , sketch the graph of a function that has the properties described. f ( x ) defined only for x ≥ 0 , ( 0 , 0 ) and ( 5 , 6 ) are on the graph; f ' ( x ) > 0 for x ≥ 0 ; f ' ' ( x ) < 0 for x < 5 , f ' ' ( 5 ) = 0 , f ' ' ( x ) > 0 for x > 5 .
In Exercises 7 − 12 , sketch the graph of a function that has the properties described. f ( x ) defined only for x ≥ 0 , ( 0 , 0 ) and ( 5 , 6 ) are on the graph; f ' ( x ) > 0 for x ≥ 0 ; f ' ' ( x ) < 0 for x < 5 , f ' ' ( 5 ) = 0 , f ' ' ( x ) > 0 for x > 5 .
Solution Summary: The author illustrates the graph of the function f(x) using the first and second derivative rule and the given properties.
In Exercises
7
−
12
, sketch the graph of a function that has the properties described.
f
(
x
)
defined only for
x
≥
0
,
(
0
,
0
)
and
(
5
,
6
)
are on the graph;
f
'
(
x
)
>
0
for
x
≥
0
;
f
'
'
(
x
)
<
0
for
x
<
5
,
f
'
'
(
5
)
=
0
,
f
'
'
(
x
)
>
0
for
x
>
5
.
4. Suppose that the population of a certain collection of rare Brazilian ants is given by
P(t)=(t+100) In(t+2),
Where t represents the time in days. Find and interpret the rates of change of the population on the third day
and on the tenth day.
Find all values of x for f (x)=(x²-4) 4 where the tangent line is horizontal.
5. Find the slope of the tangent line to the graph of f(x)=-√8x+1 at x=1. Write the equation of the tangent
line.
3. Find the derivative of each function. Label with appropriate derivative notation showing both dependent and
independent variables.
f(t)=4t(2t⭑+4)³
a. f(t)=4t (2t+4)³ (Answer must be factored.)
b.
y=
3
1
(2x³-4)
6
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY