Question 1

Which of the following best describes the tangent line to a curve at a given point?

Accepted Answer

A line that touches the curve at only one point and has the same slope as the curve at that point.

Question 2

Given the formula for the slope of a tangent line at a point as $$ 	ext{slope of tangent} = \frac{	ext{rise}}{	ext{run}} $$, what does 'rise' represent in the context of a graph?

Accepted Answer

The vertical distance between two points.

Question 3

If a tangent line at point (2, 4) on a curve passes through (2, 4) and (4, 8), what is the slope of the tangent line?

Accepted Answer

$$ \frac{8-4}{4-2} = 2 $$

Question 4

Why is the tangent line important in microeconomics when analyzing cost or revenue curves?

Accepted Answer

It indicates the instantaneous rate of change, such as marginal cost or marginal revenue.

Question 5

Suppose the tangent at point A on a total cost curve has a slope of 3. What does this value represent?

Accepted Answer

The marginal cost at point A.

Question 6

If the slope of the tangent line to a total revenue curve at a certain quantity is zero, what does this imply about marginal revenue at that quantity?

Accepted Answer

Marginal revenue is zero.

Question 7

A student draws a tangent line at point (3, 5) on a curve and calculates the slope as $$ \frac{5-2}{3-1} . $$What mistake did the student most likely make?

Accepted Answer

Used a secant line instead of a tangent line.

Question 8

Which of the following best explains why the tangent line only touches the curve at one point?

Accepted Answer

Because it represents the instantaneous rate of change at that point.

Question 9

If the tangent line at a point on a demand curve has a negative slope, what does this indicate about the relationship between price and quantity demanded at that point?

Accepted Answer

As price increases, quantity demanded decreases.

Question 10

A tangent line is drawn at point (x, y) on a curve. If the slope of the tangent is calculated as $$ \frac{\Delta y}{\Delta x} $$, what does this value represent in economic analysis?

Accepted Answer

The marginal change in y with respect to x at that point.

Slope of a Curve at a Point: Tangent and Point Method

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