Lesson test: Two problems algebra cannot solve

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Two problems algebra cannot solve

1.[2p]
Why does the two-point slope formula fail to give the steepness of a curve at a single point?

Both the rise and the run are zero, so the ratio is $0 / 0$ and is not a number

Correct

Curves have no slope anywhere, only tangent lines do

The answer is: Both the rise and the run are zero, so the ratio is $0/0$ and is not a number

The formula only works for straight lines because slopes must be constant

The answer is: Both the rise and the run are zero, so the ratio is $0/0$ and is not a number

The point may not lie on the curve

The answer is: Both the rise and the run are zero, so the ratio is $0/0$ and is not a number

2.[2p]
For $y = x^{2}$ , what is the slope of the secant joining the points at $x = 1$ and $x = 1.01$ ?

CorrectNot quite: 2.01

3.[2p]
Using four strips of equal width and the height of $y = x^{2}$ at the right end of each, what is the estimated area under the curve from $x = 0$ to $x = 1$ ?

CorrectNot quite: 0.46875

4.[1p]
For a function that increases across the whole interval, a left-endpoint rectangle sum underestimates the area.

True

Correct

False

The answer is: True

5.[2p]
Archimedes found the area of a parabolic segment exactly. What did his construction sum?

A geometric series in which each generation of inscribed triangles has a quarter of the area of the previous one

Correct

The areas of a hundred thin rectangles under the curve

The answer is: A geometric series in which each generation of inscribed triangles has a quarter of the area of the previous one

An arithmetic series of chord lengths

The answer is: A geometric series in which each generation of inscribed triangles has a quarter of the area of the previous one

The perimeters of inscribed and circumscribed polygons

The answer is: A geometric series in which each generation of inscribed triangles has a quarter of the area of the previous one

6.[3p]
A body falls from rest with speed $v = 9.8 t$ metres per second. Using the area under that graph, how far does it fall in the first 4 seconds, in metres?

CorrectNot quite: 78.4

7.[2p]
Match each operation on a motion graph to what it produces.

Slope of a position graph
Area under a velocity graph
Slope of a velocity graph
Area under an acceleration graph

distance travelled
acceleration
change in velocity
velocity

Show the answer

Slope of a position graph: velocity Area under a velocity graph: distance travelled Slope of a velocity graph: acceleration Area under an acceleration graph: change in velocity

8.[2p]
What was Berkeley's objection in The Analyst of 1734?

An increment is divided by, which requires it to be non-zero, and then discarded, which requires it to be zero

Correct

The results contradicted the measurements of astronomers

The answer is: An increment is divided by, which requires it to be non-zero, and then discarded, which requires it to be zero

Newton and Leibniz had reached different answers for the same problems

The answer is: An increment is divided by, which requires it to be non-zero, and then discarded, which requires it to be zero

Infinite series cannot have finite sums

The answer is: An increment is divided by, which requires it to be non-zero, and then discarded, which requires it to be zero

9.[2p]
Put these in the order they happened.

Cauchy and Weierstrass rebuild the subject on limits
Berkeley attacks the logic of infinitesimals
Archimedes computes the area of a parabolic segment by exhaustion
Newton and Leibniz publish a calculus of rules

Show the answer

a, b, c, d

1.[2p] Why does the two-point slope formula fail to give the steepness of a curve at a single point?

2.[2p] For y=x2, what is the slope of the secant joining the points at x=1 and x=1.01?

3.[2p] Using four strips of equal width and the height of y=x2 at the right end of each, what is the estimated area under the curve from x=0 to x=1?

4.[1p] For a function that increases across the whole interval, a left-endpoint rectangle sum underestimates the area.

5.[2p] Archimedes found the area of a parabolic segment exactly. What did his construction sum?

6.[3p] A body falls from rest with speed v=9.8t metres per second. Using the area under that graph, how far does it fall in the first 4 seconds, in metres?

7.[2p] Match each operation on a motion graph to what it produces.

8.[2p] What was Berkeley's objection in The Analyst of 1734?

9.[2p] Put these in the order they happened.

1.[2p]
Why does the two-point slope formula fail to give the steepness of a curve at a single point?

2.[2p]
For $y = x^{2}$ , what is the slope of the secant joining the points at $x = 1$ and $x = 1.01$ ?

3.[2p]
Using four strips of equal width and the height of $y = x^{2}$ at the right end of each, what is the estimated area under the curve from $x = 0$ to $x = 1$ ?

4.[1p]
For a function that increases across the whole interval, a left-endpoint rectangle sum underestimates the area.

5.[2p]
Archimedes found the area of a parabolic segment exactly. What did his construction sum?

6.[3p]
A body falls from rest with speed $v = 9.8 t$ metres per second. Using the area under that graph, how far does it fall in the first 4 seconds, in metres?

7.[2p]
Match each operation on a motion graph to what it produces.

8.[2p]
What was Berkeley's objection in The Analyst of 1734?

9.[2p]
Put these in the order they happened.