CALCULUS EXAM QUESTIONS WITH
100% CORRECT ANSWERS
For a line of slope m through the point (x0, y0): - ANSWER-Slope=m=y-y0/x-x0, and in
point-slope form:
y-y0=m(x-x0)
If y is a function of t, so y=ƒ(t), then the Average Rate of Change of y= - ANSWER-
=∆y/∆t=ƒ(b)-ƒ(a)/b-a, between t=a and t=b
The units of average rate of change of a function are units of y per unit of t
The graph of a function is __________ ___________ if it bends upward as we move left
to right - ANSWER-concave up
The graph of a function is _________ ___________ if it bends downward as we move
from left to right - ANSWER-concave down
Average velocity: - ANSWER-=Change in distance/Change in time=∆y/∆t= Average rate
of change of distance with respect to time
Relative change: - ANSWER-EX-Is a population increase of 1,000 a significant change
in a city? It depends on the original size of the community. If a town of 1,559 increases
by 1,000 people, the townspeople would definitely notice. If New York City, population
8.5 million increased by 1,000 people, almost no one will notice. To visualize the impact
of the increase on two different communities we look at the change, 1,000 as a fraction
or percentage, of the initial population. This percent change is called the relative change
In general, when a quantity P changes from P0 to P1, we define Relative Change in P
as: - ANSWER-P=Change in P/P0=P1-P0/P0
The relative change is a number without units. It is often expressed as a percentage
Cost Function: - ANSWER-C(q), gives the total cost of producing a quantity q of some
good
Costs of production can be separated into two parts: - ANSWER--Fixed costs, which are
incurred even if nothing is produced
-Variable costs, which depend on how many units are produced
*The variable cost for one additional unit is called the marginal cost
If C(q), is a linear cost function: - ANSWER--Fixed costs are represented by the vertical
intercept
-Marginal cost is represented by the slope
, Revenue Function: - ANSWER-R(q), gives the total revenue received by a firm from
selling a quantity, q, of some good. If the good sells for a prince of p per unit, and the
quantity sold is q, then:
Revenue=Price*Quantity, so R=pq
If the price does not depend on the quantity sold, so p is a constant, the graph of
revenue as a function of q is a line through the origin, with slope equal to the price p
Profit Function: - ANSWER-Profit=Revenue-Cost so π=R-C
Break-even point for a company: - ANSWER-Is the point where the profit is zero and
revenue equals cost
Supply Curve: - ANSWER-For a given item, relates the quantity, q, of the item that
manufacturers are willing to make per unit time to the price, p for which the item can be
sold. Price is the independent variable and quantity is the dependent variable
Demand Curve: - ANSWER-Relates the quantity q, of an item demanded by consumers
per unit time to the price, p of the item. Price is the independent variable, and quantity is
the dependent variable
Equilibrium point: - ANSWER-If we plot the supply and demand curves on the same
axis, the graphs cross at the equilibrium point. The values p* and q* are called the
equilibrium price and equilibrium quantity, respectively. It is assumed that the market
naturally settles to this equilibrium point
Specific tax: - ANSWER-Is a fixed amount per unit of a product sold regardless of the
selling price
Sales tax: - ANSWER-Is a fixed percentage of the selling price
Exponential Function: - ANSWER-ƒ(x)=2^x, where the power variable, is an exponential
function and the number 2 is called the base.
In the form ƒ(x)=k*a^x, where a is a positive constant, are used to represent many
phenomena in the natural and social sciences
For a Population Growth: - ANSWER-The base of the equation that is raised to the
exponent (the exponential function) is the factor by which the population grows each
year and is called the growth factor
Exponential decay function: - ANSWER-When the base of the function that is raised to
the exponent(the exponential function) is less than one it is an exponential decay
function because as the exponent the base is being raised to increases, the function
values get closer to zero
P is an exponential function of t with base a if: - ANSWER-P=P0a^t, where P0 is the
initial quantity(when t=0) and a if the factor by which P changes when t increases by 1.
100% CORRECT ANSWERS
For a line of slope m through the point (x0, y0): - ANSWER-Slope=m=y-y0/x-x0, and in
point-slope form:
y-y0=m(x-x0)
If y is a function of t, so y=ƒ(t), then the Average Rate of Change of y= - ANSWER-
=∆y/∆t=ƒ(b)-ƒ(a)/b-a, between t=a and t=b
The units of average rate of change of a function are units of y per unit of t
The graph of a function is __________ ___________ if it bends upward as we move left
to right - ANSWER-concave up
The graph of a function is _________ ___________ if it bends downward as we move
from left to right - ANSWER-concave down
Average velocity: - ANSWER-=Change in distance/Change in time=∆y/∆t= Average rate
of change of distance with respect to time
Relative change: - ANSWER-EX-Is a population increase of 1,000 a significant change
in a city? It depends on the original size of the community. If a town of 1,559 increases
by 1,000 people, the townspeople would definitely notice. If New York City, population
8.5 million increased by 1,000 people, almost no one will notice. To visualize the impact
of the increase on two different communities we look at the change, 1,000 as a fraction
or percentage, of the initial population. This percent change is called the relative change
In general, when a quantity P changes from P0 to P1, we define Relative Change in P
as: - ANSWER-P=Change in P/P0=P1-P0/P0
The relative change is a number without units. It is often expressed as a percentage
Cost Function: - ANSWER-C(q), gives the total cost of producing a quantity q of some
good
Costs of production can be separated into two parts: - ANSWER--Fixed costs, which are
incurred even if nothing is produced
-Variable costs, which depend on how many units are produced
*The variable cost for one additional unit is called the marginal cost
If C(q), is a linear cost function: - ANSWER--Fixed costs are represented by the vertical
intercept
-Marginal cost is represented by the slope
, Revenue Function: - ANSWER-R(q), gives the total revenue received by a firm from
selling a quantity, q, of some good. If the good sells for a prince of p per unit, and the
quantity sold is q, then:
Revenue=Price*Quantity, so R=pq
If the price does not depend on the quantity sold, so p is a constant, the graph of
revenue as a function of q is a line through the origin, with slope equal to the price p
Profit Function: - ANSWER-Profit=Revenue-Cost so π=R-C
Break-even point for a company: - ANSWER-Is the point where the profit is zero and
revenue equals cost
Supply Curve: - ANSWER-For a given item, relates the quantity, q, of the item that
manufacturers are willing to make per unit time to the price, p for which the item can be
sold. Price is the independent variable and quantity is the dependent variable
Demand Curve: - ANSWER-Relates the quantity q, of an item demanded by consumers
per unit time to the price, p of the item. Price is the independent variable, and quantity is
the dependent variable
Equilibrium point: - ANSWER-If we plot the supply and demand curves on the same
axis, the graphs cross at the equilibrium point. The values p* and q* are called the
equilibrium price and equilibrium quantity, respectively. It is assumed that the market
naturally settles to this equilibrium point
Specific tax: - ANSWER-Is a fixed amount per unit of a product sold regardless of the
selling price
Sales tax: - ANSWER-Is a fixed percentage of the selling price
Exponential Function: - ANSWER-ƒ(x)=2^x, where the power variable, is an exponential
function and the number 2 is called the base.
In the form ƒ(x)=k*a^x, where a is a positive constant, are used to represent many
phenomena in the natural and social sciences
For a Population Growth: - ANSWER-The base of the equation that is raised to the
exponent (the exponential function) is the factor by which the population grows each
year and is called the growth factor
Exponential decay function: - ANSWER-When the base of the function that is raised to
the exponent(the exponential function) is less than one it is an exponential decay
function because as the exponent the base is being raised to increases, the function
values get closer to zero
P is an exponential function of t with base a if: - ANSWER-P=P0a^t, where P0 is the
initial quantity(when t=0) and a if the factor by which P changes when t increases by 1.
