Lesson test: The Fundamental Theorem

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The Fundamental Theorem

1.[2p]
Evaluate $\int_{0}^{π} \sin x d x$ .

CorrectNot quite: 2

2.[2p]
Evaluate $\int_{0}^{2} (3 x^{2} - 2 x) d x$ .

CorrectNot quite: 4

3.[3p]
Part two of the theorem works with any antiderivative $F$ . What guarantees that the choice does not matter?

Two antiderivatives differ by a constant, which cancels in the subtraction $F (b) - F (a)$

Correct

There is only one antiderivative of a continuous function

The answer is: Two antiderivatives differ by a constant, which cancels in the subtraction $F (b) - F (a)$

The constant is always zero

The answer is: Two antiderivatives differ by a constant, which cancels in the subtraction $F (b) - F (a)$

The integral is defined as $F (b) - F (a)$

The answer is: Two antiderivatives differ by a constant, which cancels in the subtraction $F (b) - F (a)$

4.[3p]
If $G (x) = \int_{1}^{x^{3}} \ln t d t$ , find $G' (2)$ , to two decimal places.

CorrectNot quite: 24.95

5.[3p]
Evaluating $\int_{- 1}^{1} d x / x^{2}$ as $[- 1 / x]$ gives $- 2$ , which is impossible for a positive integrand. Why?

The integrand is unbounded at $x = 0$ , inside the interval, so the theorem does not apply

Correct

The antiderivative was wrong

The answer is: The integrand is unbounded at $x = 0$ , inside the interval, so the theorem does not apply

The limits were taken in the wrong order

The answer is: The integrand is unbounded at $x = 0$ , inside the interval, so the theorem does not apply

The integral of an even function must be zero

The answer is: The integrand is unbounded at $x = 0$ , inside the interval, so the theorem does not apply

6.[2p]
Water enters a tank at $r (t) = 200 + 20 t$ litres per minute. How many litres enter in the first 10 minutes?

CorrectNot quite: 3000

7.[3p]
A particle has velocity $v (t) = t^{2} - 4$ m/s. What is its displacement over the first 3 seconds, in metres? Give the signed value.

CorrectNot quite: -3

8.[2p]
Every continuous function has an antiderivative, though it may not be expressible in elementary terms.

True

Correct

False

The answer is: True

9.[3p]
Match each integrand to its antiderivative.

$x^{n}$ with $n \neq - 1$
$1 / x$
$\cos x$
$1 / (1 + x^{2})$

$\arctan x$
$x^{n + 1} / (n + 1)$
$\ln | x |$
$\sin x$

Show the answer

$x^{n}$ with $n \neq - 1$ : $x^{n + 1} / (n + 1)$ $1 / x$ : $\ln | x |$ $\cos x$ : $\sin x$ $1 / (1 + x^{2})$ : $\arctan x$

1.[2p] Evaluate ∫0πsinxdx.

2.[2p] Evaluate ∫02(3x2−2x)dx.

3.[3p] Part two of the theorem works with any antiderivative F. What guarantees that the choice does not matter?

4.[3p] If G(x)=∫1x3lntdt, find G′(2), to two decimal places.

5.[3p] Evaluating ∫−11dx/x2 as [−1/x] gives −2, which is impossible for a positive integrand. Why?

6.[2p] Water enters a tank at r(t)=200+20t litres per minute. How many litres enter in the first 10 minutes?

7.[3p] A particle has velocity v(t)=t2−4 m/s. What is its displacement over the first 3 seconds, in metres? Give the signed value.

8.[2p] Every continuous function has an antiderivative, though it may not be expressible in elementary terms.

9.[3p] Match each integrand to its antiderivative.

1.[2p]
Evaluate $\int_{0}^{π} \sin x d x$ .

2.[2p]
Evaluate $\int_{0}^{2} (3 x^{2} - 2 x) d x$ .

3.[3p]
Part two of the theorem works with any antiderivative $F$ . What guarantees that the choice does not matter?

4.[3p]
If $G (x) = \int_{1}^{x^{3}} \ln t d t$ , find $G' (2)$ , to two decimal places.

5.[3p]
Evaluating $\int_{- 1}^{1} d x / x^{2}$ as $[- 1 / x]$ gives $- 2$ , which is impossible for a positive integrand. Why?

6.[2p]
Water enters a tank at $r (t) = 200 + 20 t$ litres per minute. How many litres enter in the first 10 minutes?

7.[3p]
A particle has velocity $v (t) = t^{2} - 4$ m/s. What is its displacement over the first 3 seconds, in metres? Give the signed value.

8.[2p]
Every continuous function has an antiderivative, though it may not be expressible in elementary terms.

9.[3p]
Match each integrand to its antiderivative.