Problems on tangents, velocity, derivatives, and differentiation.
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Transcript of Problems on tangents, velocity, derivatives, and differentiation.
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problems on tangents, velocity, derivatives, and
differentiation
![Page 2: Problems on tangents, velocity, derivatives, and differentiation.](https://reader035.fdocuments.net/reader035/viewer/2022062307/55154ebc55034673228b63b0/html5/thumbnails/2.jpg)
Mika Seppälä: Problems on Tangents, Velocity, Derivatives, and Differentiation
overview of problemsA rock is thrown upward with a velocity of
10m/s, its height in meters t seconds later
is given by . y =10t −1.86t2
1
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Mika Seppälä: Problems on Tangents, Velocity, Derivatives, and Differentiation
overview of problems1-a
1-b Estimate the instantaneous velocity
when . t =3
Find the average velocity over the given
time intervals.
(i) (ii) (iii) 1,1.1⎡⎣
⎤⎦
1,1.001⎡⎣
⎤⎦
1,1.01⎡⎣
⎤⎦
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Mika Seppälä: Problems on Tangents, Velocity, Derivatives, and Differentiation
overview of problemsThe table shows the position of a cyclist.
t(seconds) 0 1 2 3 4 5
s(meters) 0 1.4 5.1 10.7 17.7 25.8
2
2-a
2-b Estimate the instantaneous velocity
when . t =3
Find the average velocity for each time
period.
(i) (ii) (iii) 1,3⎡⎣
⎤⎦
2,3⎡⎣
⎤⎦
3,5⎡⎣
⎤⎦
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Mika Seppälä: Problems on Tangents, Velocity, Derivatives, and Differentiation
overview of problems3 The graph of a
function and that of
its derivative is
given.
Which is which?
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Mika Seppälä: Problems on Tangents, Velocity, Derivatives, and Differentiation
overview of problemsFind the derivative of the functions below
using the definition of the derivative.
g x( ) =
1x
u x( ) =
1x2
h x( ) = 1 + x
v x( ) = 1 −x( )
−1 2
f x( ) =2 x3 −3x + 5
4
4-a
4-c
4-e
4-b
4-d
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Mika Seppälä: Problems on Tangents, Velocity, Derivatives, and Differentiation
overview of problemsState, with reasons, the numbers at
which f is not differentiable.
5
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Mika Seppälä: Problems on Tangents, Velocity, Derivatives, and Differentiation
overview of problemsFind the equation of the line tangent to
the graph of the function at the
point .
y =x3
1,1( )
Find all points on the graph of the
function at which the
tangent line is horizontal. y =sin x( ) −cos x( )
6
7
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Mika Seppälä: Problems on Tangents, Velocity, Derivatives, and Differentiation
overview of problems
f x( ) =
x sin 1 x( ) , x ≠0
0 , x =0
⎧⎨⎪
⎩⎪
g x( ) =
x2 sin 1 x( ) , x ≠0
0 , x =0
⎧⎨⎪
⎩⎪
8
8-a
8-b
Do the following functions have
derivative at ? x =0
8-c h x( ) =x x
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Mika Seppälä: Problems on Tangents, Velocity, Derivatives, and Differentiation
overview of problems9
10
Assume that f has derivative everywhere.
Set . Using the definition of
derivative, show that g has a derivative
and that .
g x( ) =xf x( )
′g x( ) =f x( ) + x ′f x( )
Show that the function is
differentiable everywhere. f x( ) =x2
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Mika Seppälä: Problems on Tangents, Velocity, Derivatives, and Differentiation
overview of problems11
f x( ) =x2 cos
1x
⎛
⎝⎜⎞
⎠⎟, x ≠0
0 , x =0
⎧
⎨⎪
⎩⎪
Show that f is differentiable at . x =0