Lesson test: Implicit differentiation and related rates

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Implicit differentiation and related rates

1.[2p]
For the circle $x^{2} + y^{2} = 25$ , what is $d y / d x$ at the point $(3, 4)$ ?

CorrectNot quite: -0.75

2.[3p]
For the curve $x^{2} + x y + y^{2} = 7$ , what is $d y / d x$ at the point $(1, 2)$ ?

CorrectNot quite: -0.8

3.[2p]
Why is implicit differentiation needed for a curve such as $x^{3} + y^{3} = 6 x y$ ?

No usable formula for $y$ in terms of $x$ exists, and the equation can be differentiated without one

Correct

The curve is not continuous

The answer is: No usable formula for $y$ in terms of $x$ exists, and the equation can be differentiated without one

The chain rule does not apply to cubes

The answer is: No usable formula for $y$ in terms of $x$ exists, and the equation can be differentiated without one

The curve has no tangent lines

The answer is: No usable formula for $y$ in terms of $x$ exists, and the equation can be differentiated without one

4.[2p]
What is the derivative of $\arctan x$ at $x = 1$ ?

CorrectNot quite: 0.5

5.[3p]
A 10 m ladder leans on a wall and its foot is pulled out at 2 m/s. How fast is the top descending when the foot is 6 m from the wall, in metres per second? Give the signed value.

CorrectNot quite: -1.5

6.[3p]
Air enters a spherical balloon at 100 cm³ per second. How fast is the radius growing when the radius is 5 cm, in cm per second?

CorrectNot quite: 0.318

7.[3p]
What is the standard error in a related rates problem?

Substituting the values that hold at one instant before differentiating, which deletes the terms carrying the rates

Correct

Differentiating with respect to the wrong variable

The answer is: Substituting the values that hold at one instant before differentiating, which deletes the terms carrying the rates

Using similar triangles instead of Pythagoras

The answer is: Substituting the values that hold at one instant before differentiating, which deletes the terms carrying the rates

Forgetting to take the square root at the end

The answer is: Substituting the values that hold at one instant before differentiating, which deletes the terms carrying the rates

8.[3p]
Match each inverse function to its derivative.

$\arcsin x$
$\arctan x$
$\ln x$
$x^{1 / 2}$

${\frac{12}{x}}^{- 1 / 2}$
$1 / (1 + x^{2})$
$1 / \sqrt{1 - x^{2}}$
$1 / x$

Show the answer

$\arcsin x$ : $1 / \sqrt{1 - x^{2}}$ $\arctan x$ : $1 / (1 + x^{2})$ $\ln x$ : $1 / x$ $x^{1 / 2}$ : ${\frac{12}{x}}^{- 1 / 2}$

9.[2p]
Implicit differentiation is what extends the power rule from whole-number exponents to fractional ones.

True

Correct

False

The answer is: True

1.[2p] For the circle x2+y2=25, what is dy/dx at the point (3,4)?

2.[3p] For the curve x2+xy+y2=7, what is dy/dx at the point (1,2)?

3.[2p] Why is implicit differentiation needed for a curve such as x3+y3=6xy?

4.[2p] What is the derivative of arctanx at x=1?

5.[3p] A 10 m ladder leans on a wall and its foot is pulled out at 2 m/s. How fast is the top descending when the foot is 6 m from the wall, in metres per second? Give the signed value.

6.[3p] Air enters a spherical balloon at 100 cm³ per second. How fast is the radius growing when the radius is 5 cm, in cm per second?

7.[3p] What is the standard error in a related rates problem?

8.[3p] Match each inverse function to its derivative.

9.[2p] Implicit differentiation is what extends the power rule from whole-number exponents to fractional ones.

1.[2p]
For the circle $x^{2} + y^{2} = 25$ , what is $d y / d x$ at the point $(3, 4)$ ?

2.[3p]
For the curve $x^{2} + x y + y^{2} = 7$ , what is $d y / d x$ at the point $(1, 2)$ ?

3.[2p]
Why is implicit differentiation needed for a curve such as $x^{3} + y^{3} = 6 x y$ ?

4.[2p]
What is the derivative of $\arctan x$ at $x = 1$ ?

5.[3p]
A 10 m ladder leans on a wall and its foot is pulled out at 2 m/s. How fast is the top descending when the foot is 6 m from the wall, in metres per second? Give the signed value.

6.[3p]
Air enters a spherical balloon at 100 cm³ per second. How fast is the radius growing when the radius is 5 cm, in cm per second?

7.[3p]
What is the standard error in a related rates problem?

8.[3p]
Match each inverse function to its derivative.

9.[2p]
Implicit differentiation is what extends the power rule from whole-number exponents to fractional ones.