Lesson test: The derivative

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The derivative

1.[2p]
Using the definition, find the derivative of $f (x) = x^{2} - 5 x$ at $x = 1$ .

CorrectNot quite: -3

2.[3p]
Find the derivative of $f (x) = \sqrt{x}$ at $x = 9$ , to four decimal places.

CorrectNot quite: 0.1667

3.[2p]
A stone falls with distance $s = 4.9 t^{2}$ metres. What is its speed at $t = 3$ seconds, in metres per second?

CorrectNot quite: 29.4

4.[2p]
Why must the difference quotient be evaluated as a limit rather than by substitution?

At $h = 0$ it is $0 / 0$ , which is undefined, while every other $h$ gives a secant rather than a tangent

Correct

Because substitution is only valid for polynomials

The answer is: At $h = 0$ it is $0/0$, which is undefined, while every other $h$ gives a secant rather than a tangent

Because $f (a + h)$ cannot be computed when $h$ is small

The answer is: At $h = 0$ it is $0/0$, which is undefined, while every other $h$ gives a secant rather than a tangent

Because the tangent line has no slope

The answer is: At $h = 0$ it is $0/0$, which is undefined, while every other $h$ gives a secant rather than a tangent

5.[2p]
A function that is differentiable at a point is continuous there.

True

Correct

False

The answer is: True

6.[2p]
Which function is continuous at $x = 0$ but has no derivative there?

$f (x) = | x |$

Correct

$f (x) = x^{2}$

The answer is: $f(x) = |x|$

$f (x) = 1 / x$

The answer is: $f(x) = |x|$

$f (x) = 3 x + 1$

The answer is: $f(x) = |x|$

7.[3p]
In which of these ways can a derivative fail to exist at a point?

Select all that apply

The one-sided limits of the difference quotient disagree, giving a corner

Correct

The difference quotient grows without bound from both sides, giving a vertical tangent

Correct

The function has a jump there

Correct

The function is not a polynomial

The answer is: The one-sided limits of the difference quotient disagree, giving a corner, The difference quotient grows without bound from both sides, giving a vertical tangent, The function has a jump there

8.[2p]
Match each notation to what it denotes.

$f^{'} (x)$
$d y / d x$
$f^{''} (x)$
$\ddot{x}$

the rate of change of the slope
the derivative as a new function
the derivative naming both variables
the second derivative with respect to time

Show the answer

$f^{'} (x)$ : the derivative as a new function $d y / d x$ : the derivative naming both variables $f^{''} (x)$ : the rate of change of the slope $\ddot{x}$ : the second derivative with respect to time

9.[2p]
Put the steps of differentiating from the definition in order.

Expand and simplify the numerator
Write the difference quotient with $f (x + h)$ and $f (x)$
Take the limit as $h$ approaches zero
Cancel the factor of $h$ that every term carries

Show the answer

a, b, c, d

1.[2p] Using the definition, find the derivative of f(x)=x2-5x at x=1.

2.[3p] Find the derivative of f(x)=x at x=9, to four decimal places.

3.[2p] A stone falls with distance s=4.9t2 metres. What is its speed at t=3 seconds, in metres per second?

4.[2p] Why must the difference quotient be evaluated as a limit rather than by substitution?

5.[2p] A function that is differentiable at a point is continuous there.

6.[2p] Which function is continuous at x=0 but has no derivative there?

7.[3p] In which of these ways can a derivative fail to exist at a point?

8.[2p] Match each notation to what it denotes.

9.[2p] Put the steps of differentiating from the definition in order.

1.[2p]
Using the definition, find the derivative of $f (x) = x^{2} - 5 x$ at $x = 1$ .

2.[3p]
Find the derivative of $f (x) = \sqrt{x}$ at $x = 9$ , to four decimal places.

3.[2p]
A stone falls with distance $s = 4.9 t^{2}$ metres. What is its speed at $t = 3$ seconds, in metres per second?

4.[2p]
Why must the difference quotient be evaluated as a limit rather than by substitution?

5.[2p]
A function that is differentiable at a point is continuous there.

6.[2p]
Which function is continuous at $x = 0$ but has no derivative there?

7.[3p]
In which of these ways can a derivative fail to exist at a point?

8.[2p]
Match each notation to what it denotes.

9.[2p]
Put the steps of differentiating from the definition in order.