Lesson test: Equations for a function

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Equations for a function

1.[2p]
What makes a differential equation different from an algebraic equation?

Its unknown is a function, so a solution is a whole curve rather than a number

Correct

It always has infinitely many numerical roots

The answer is: Its unknown is a function, so a solution is a whole curve rather than a number

It can only be checked, never solved

The answer is: Its unknown is a function, so a solution is a whole curve rather than a number

Its unknown is a number defined by a limit

The answer is: Its unknown is a function, so a solution is a whole curve rather than a number

2.[2p]
For the direction field of $y' = t - y$ , what is the slope of the solution passing through the point $t = 3$ , $y = 1$ ?

CorrectNot quite: 2

3.[2p]
Which of these equations is linear?

$y' + t^{2} y = \cos t$

Correct

$y' = y^{2}$

The answer is: $y' + t^2 y = \cos t$

$y'' + \sin y = 0$

The answer is: $y' + t^2 y = \cos t$

$y y' = t$

The answer is: $y' + t^2 y = \cos t$

4.[1p]
If $y_{1}$ and $y_{2}$ each solve $y' = y^{2}$ , then $y_{1} + y_{2}$ also solves it.

True

The answer is: False

False

Correct

5.[3p]
Carbon-14 has a half-life of 5730 years. A sample holds 40 per cent of its original carbon-14. How old is it, in years?

CorrectNot quite: 7575

6.[3p]
The general solution of $y'' + 4 y = 0$ is $y = A \cos 2 t + B \sin 2 t$ . For $y (0) = 5$ and $y' (0) = 24$ , what is the amplitude $\sqrt{A^{2} + B^{2}}$ ?

CorrectNot quite: 13

7.[3p]
Match each equation to its order and linearity.

$y' = - k N$
$y y' = t$
$m x'' + c x' + k x = 0$
$θ'' + \sin θ = 0$

first order, linear
second order, nonlinear
second order, linear
first order, nonlinear

Show the answer

$y' = - k N$ : first order, linear $y y' = t$ : first order, nonlinear $m x'' + c x' + k x = 0$ : second order, linear $θ'' + \sin θ = 0$ : second order, nonlinear

8.[3p]
Which statements about a direction field are correct?

Select all that apply

It gives the slope of a solution at every point without any solving

Correct

A curve on which the slope takes one fixed value is called an isocline

Correct

Solutions are the curves that stay tangent to it everywhere

Correct

It can only be drawn once the general solution is known

The answer is: It gives the slope of a solution at every point without any solving, A curve on which the slope takes one fixed value is called an isocline, Solutions are the curves that stay tangent to it everywhere

9.[2p]
A body falls under gravity with drag, obeying $m d v / d t = m g - b v^{2}$ . What can be read off the equation before solving it?

The speed stops changing once $b v^{2}$ reaches $m g$ , giving a terminal speed of $\sqrt{m g / b}$

Correct

The speed grows without bound, since gravity always acts

The answer is: The speed stops changing once $bv^2$ reaches $mg$, giving a terminal speed of $\sqrt{mg/b}$

The solution is a sine, because the equation is second order

The answer is: The speed stops changing once $bv^2$ reaches $mg$, giving a terminal speed of $\sqrt{mg/b}$

Nothing at all until the equation is integrated

The answer is: The speed stops changing once $bv^2$ reaches $mg$, giving a terminal speed of $\sqrt{mg/b}$

1.[2p] What makes a differential equation different from an algebraic equation?

2.[2p] For the direction field of y'=t-y, what is the slope of the solution passing through the point t=3, y=1?

3.[2p] Which of these equations is linear?

4.[1p] If y1 and y2 each solve y'=y2, then y1+y2 also solves it.

5.[3p] Carbon-14 has a half-life of 5730 years. A sample holds 40 per cent of its original carbon-14. How old is it, in years?

6.[3p] The general solution of y''+4y=0 is y=Acos2t+Bsin2t. For y(0)=5 and y'(0)=24, what is the amplitude A2+B2?

7.[3p] Match each equation to its order and linearity.

8.[3p] Which statements about a direction field are correct?

9.[2p] A body falls under gravity with drag, obeying mdv/dt=mg-bv2. What can be read off the equation before solving it?

1.[2p]
What makes a differential equation different from an algebraic equation?

2.[2p]
For the direction field of $y' = t - y$ , what is the slope of the solution passing through the point $t = 3$ , $y = 1$ ?

3.[2p]
Which of these equations is linear?

4.[1p]
If $y_{1}$ and $y_{2}$ each solve $y' = y^{2}$ , then $y_{1} + y_{2}$ also solves it.

5.[3p]
Carbon-14 has a half-life of 5730 years. A sample holds 40 per cent of its original carbon-14. How old is it, in years?

6.[3p]
The general solution of $y'' + 4 y = 0$ is $y = A \cos 2 t + B \sin 2 t$ . For $y (0) = 5$ and $y' (0) = 24$ , what is the amplitude $\sqrt{A^{2} + B^{2}}$ ?

7.[3p]
Match each equation to its order and linearity.

8.[3p]
Which statements about a direction field are correct?

9.[2p]
A body falls under gravity with drag, obeying $m d v / d t = m g - b v^{2}$ . What can be read off the equation before solving it?