problems on tangents, velocity, derivatives, and

differentiation

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

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⎡⎣

⎤⎦

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⎡⎣

⎤⎦

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?

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

Mika Seppälä: Problems on Tangents, Velocity, Derivatives, and Differentiation

overview of problemsState, with reasons, the numbers at

which f is not differentiable.

5

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

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

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

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