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AP Calculus BC
Unit 1: Limits and Continuity
1.1 Introducing Calculus: Connecting Graphs and Rates of Change
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Calculus is the mathematical study of
change
Differential calculus focuses on finding the
rate of change
of a function.
Integral calculus deals with accumulating quantities and finding the area under a
curve
What does differential calculus focus on?
Rate of change
Integral calculus is the reverse process of
differentiation
.
A limit describes the value a function approaches as its input gets close to a certain
point
What is one method to find limits?
Graphically
For a limit to exist, both left and
right
limits must be equal.
As
x
→
2
x \to 2
x
→
2
, the limit of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
is 4
Match the calculus branch with its key concept:
Differential Calculus ↔️ Derivatives
Integral Calculus ↔️ Integrals
To find limits, we can use graphical or algebraic
methods
What is a one-sided limit?
Limit from one direction
A limit describes the value a function approaches as its input gets close to a certain
point
For a limit to exist, both
one-sided
limits must exist and be equal
What are one-sided limits defined as?
Values approached from one direction
A limit is notated as \lim_{x \to a} f(x) =
L
A general limit exists if both one-sided limits are
equal
The rate of change measures how a function's value changes with respect to its
input
The average rate of change is calculated using the
secant line
between two points
What does the instantaneous rate of change represent on a graph?
Tangent line at a single point
The instantaneous rate of change is equivalent to the slope of the
tangent
The instantaneous rate of change is calculated as the limit of the average rate of change as the
interval
shrinks to zero
What is the formula for the instantaneous rate of change?
\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}</latex>
What is the formula for the average rate of change?
f
(
x
+
h
)
−
f
(
x
)
h
\frac{f(x + h) - f(x)}{h}
h
f
(
x
+
h
)
−
f
(
x
)
The instantaneous rate of change of a function at a point is equivalent to the slope of the
tangent
What is the formula for the instantaneous rate of change using limits?
\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}</latex>
For
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
, the slope of the tangent line at
x
=
x =
x
=
3
3
3
is 6.
Match the step with its description and formula:
Average rate of change ↔️
f
(
x
+
h
)
−
f
(
x
)
h
\frac{f(x + h) - f(x)}{h}
h
f
(
x
+
h
)
−
f
(
x
)
Simplify ↔️
6
h
+
h
2
h
\frac{6h + h^{2}}{h}
h
6
h
+
h
2
Limit as
h
→
0
h \to 0
h
→
0
↔️
lim
h
→
0
(
6
+
h
)
\lim_{h \to 0} (6 + h)
lim
h
→
0
(
6
+
h
)
What are the two main branches of calculus?
Differential and Integral
Differential calculus focuses on finding the rate of change of a function and the slopes of
curves
What is the reverse process of differentiation?
Integration
Match the branch of calculus with its key concepts:
Differential Calculus ↔️ Derivatives, Tangents
Integral Calculus ↔️ Integrals, Areas
What does the notation
lim
x
→
a
f
(
x
)
=
\lim_{x \to a} f(x) =
lim
x
→
a
f
(
x
)
=
L
L
L
mean?
As
x
x
x
approaches
a
a
a
,
f
(
x
)
f(x)
f
(
x
)
approaches
L
L
L
For a limit to exist, both the left and right limits must exist and be
equal
The limit of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
−
4
x
−
2
\frac{x^{2} - 4}{x - 2}
x
−
2
x
2
−
4
as
x
→
2
x \to 2
x
→
2
is 4.
What is the graphical interpretation of the average rate of change?
Slope of the secant line
What is the graphical interpretation of the instantaneous rate of change?
Slope of the tangent line
What is the equation of the tangent line at a point
x
=
x =
x
=
a
a
a
?
y
=
y =
y
=
f
′
(
a
)
(
x
−
a
)
+
f'(a)(x - a) +
f
′
(
a
)
(
x
−
a
)
+
f
(
a
)
f(a)
f
(
a
)
The tangent line to
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
at
x
=
x =
x
=
2
2
2
has a slope of 4.
Match the field with its application of calculus:
Economics ↔️ Profit Optimization
Physics ↔️ Motion Analysis
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