Lesson test: Sines, exponentials and their derivatives

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Sines, exponentials and their derivatives

1.[2p]
What is $lim_{h \to 0} \frac{\sin h}{h}$ , with $h$ in radians?

CorrectNot quite: 1

2.[3p]
The proof that this limit is $1$ compares the areas of a triangle, a sector and a larger triangle. Which step needs radians?

The sector area is $h / 2$ only when $h$ is the radian measure of the angle

Correct

The triangle area is $\frac{1}{2} \sin h$ only in radians

The answer is: The sector area is $h / 2$ only when $h$ is the radian measure of the angle

The squeeze theorem holds only in radians

The answer is: The sector area is $h / 2$ only when $h$ is the radian measure of the angle

The tangent function is defined only in radians

The answer is: The sector area is $h / 2$ only when $h$ is the radian measure of the angle

3.[3p]
Differentiate $\sin (3 x^{2})$ and evaluate at $x = 1$ .

CorrectNot quite: -5.94

4.[2p]
Differentiate $x e^{x}$ and evaluate at $x = 1$ .

CorrectNot quite: 5.437

5.[3p]
How is the number $e$ defined in this lesson?

As the base for which $lim_{h \to 0} (a^{h} - 1) / h$ equals $1$ , so that the exponential is its own derivative

Correct

As the sum $1 + 1 / 2 + 1 / 3 + \dots$

The answer is: As the base for which $lim_{h \to 0} (a^{h} - 1) / h$ equals $1$ , so that the exponential is its own derivative

As the value of $2.718282$ , chosen by convention

The answer is: As the base for which $lim_{h \to 0} (a^{h} - 1) / h$ equals $1$ , so that the exponential is its own derivative

As the base for which the logarithm is undefined at zero

The answer is: As the base for which $lim_{h \to 0} (a^{h} - 1) / h$ equals $1$ , so that the exponential is its own derivative

6.[2p]
What is the derivative of $2^{x}$ at $x = 0$ , to three decimal places?

CorrectNot quite: 0.693

7.[2p]
The derivative of $\ln x$ supplies the one function the power rule can never produce, the one whose derivative is $1 / x$ .

True

Correct

False

The answer is: True

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

$\sin x$
$\cos x$
$\tan x$
$\ln x$
$a^{x}$

$- \sin x$
$\sec^{2} x$
$1 / x$
$a^{x} \ln a$
$\cos x$

Show the answer

$\sin x$ : $\cos x$ $\cos x$ : $- \sin x$ $\tan x$ : $\sec^{2} x$ $\ln x$ : $1 / x$ $a^{x}$ : $a^{x} \ln a$

9.[3p]
Caffeine has a half-life of 5 hours. For a 100 mg dose, what is the elimination rate at $t = 0$ , in mg per hour? Give the signed value.

CorrectNot quite: -13.86

1.[2p] What is limh→0sinhh, with h in radians?

2.[3p] The proof that this limit is 1 compares the areas of a triangle, a sector and a larger triangle. Which step needs radians?

3.[3p] Differentiate sin(3x2) and evaluate at x=1.

4.[2p] Differentiate xex and evaluate at x=1.

5.[3p] How is the number e defined in this lesson?

6.[2p] What is the derivative of 2x at x=0, to three decimal places?

7.[2p] The derivative of lnx supplies the one function the power rule can never produce, the one whose derivative is 1/x.

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

9.[3p] Caffeine has a half-life of 5 hours. For a 100 mg dose, what is the elimination rate at t=0, in mg per hour? Give the signed value.

1.[2p]
What is $lim_{h \to 0} \frac{\sin h}{h}$ , with $h$ in radians?

2.[3p]
The proof that this limit is $1$ compares the areas of a triangle, a sector and a larger triangle. Which step needs radians?

3.[3p]
Differentiate $\sin (3 x^{2})$ and evaluate at $x = 1$ .

4.[2p]
Differentiate $x e^{x}$ and evaluate at $x = 1$ .

5.[3p]
How is the number $e$ defined in this lesson?

6.[2p]
What is the derivative of $2^{x}$ at $x = 0$ , to three decimal places?

7.[2p]
The derivative of $\ln x$ supplies the one function the power rule can never produce, the one whose derivative is $1 / x$ .

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

9.[3p]
Caffeine has a half-life of 5 hours. For a 100 mg dose, what is the elimination rate at $t = 0$ , in mg per hour? Give the signed value.