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AP Calculus BC
Unit 4: Contextual Applications of Differentiation
4.1 Interpreting the Meaning of the Derivative in Context
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The derivative represents the
slope
of the tangent line at a point.
True
What does the average rate of change represent graphically?
Slope of the secant line
The derivative of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
at
x
=
x =
x
=
3
3
3
is 6
The average rate of change is calculated using the slope of the
secant line
.
True
The tangent line touches the function at a single
point
The derivative of a
constant
function is always zero.
True
What does the derivative measure at a specific point?
Instantaneous rate of change
The derivative is calculated using the limit of the difference
quotient
The derivative represents the
slope
of the tangent line at a single point.
True
In real-world terms, what does the derivative help us understand?
Rate of change at a moment
What does a derivative value of 6 at
x
=
x =
x
=
3
3
3
indicate for
f
(
x
)
=
f(x) =
f
(
x
)
=
x
2
x^{2}
x
2
?
Function changes at a rate of 6
What is the formula for calculating the derivative using the limit of the difference quotient?
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
lim
h
→
0
h
f
(
x
+
h
)
−
f
(
x
)
The power rule states that the derivative of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
n
x^{n}
x
n
is
n
x
n
−
1
nx^{n - 1}
n
x
n
−
1
if n is any real number.
What is the derivative of
f
(
x
)
=
f(x) =
f
(
x
)
=
a
x
a^{x}
a
x
?
a
x
ln
(
a
)
a^{x} \ln(a)
a
x
ln
(
a
)
The derivative of a power function f(x) = x^{n}</latex> is
nx^{n - 1}
To apply differentiation rules, identify the function type and substitute into the corresponding
rule
What is the derivative of
f
(
x
)
=
f(x) =
f
(
x
)
=
3
x
4
+
3x^{4} +
3
x
4
+
2
e
x
−
sin
(
x
)
2e^{x} - \sin(x)
2
e
x
−
sin
(
x
)
?
12
x
3
+
12x^{3} +
12
x
3
+
2
e
x
−
cos
(
x
)
2e^{x} - \cos(x)
2
e
x
−
cos
(
x
)
Steps of the first derivative test
1️⃣ Find the critical points of the function
2️⃣ Evaluate the sign of the derivative on either side of each critical point
3️⃣ Determine if the critical point is a local maximum, minimum, or neither
If the sign of the derivative changes from positive to negative, the critical point is a local
maximum
What is the purpose of the first derivative test?
Analyze critical points
If the sign of the derivative does not change at a critical point, it is neither a local
maximum
nor a local minimum.
True
What is the formula for the derivative using the limit of the difference quotient?
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
lim
h
→
0
h
f
(
x
+
h
)
−
f
(
x
)
The derivative is essential for understanding rates of change in scenarios like determining a car's
velocity
.
True
The tangent line touches the function at a single
point
What is the instantaneous rate of change at a specific moment called?
Derivative
The derivative of a constant function is always
zero
The derivative of
a
x
a^{x}
a
x
is
a
x
ln
(
a
)
a^{x} \ln(a)
a
x
ln
(
a
)
.
True
The derivative of a constant function
f
(
x
)
=
f(x) =
f
(
x
)
=
c
c
c
is 0
What is the derivative of
f
(
x
)
=
f(x) =
f
(
x
)
=
a
x
a^{x}
a
x
?
a
x
ln
(
a
)
a^{x} \ln(a)
a
x
ln
(
a
)
The derivative of
cos
(
x
)
\cos(x)
cos
(
x
)
is
sin
(
x
)
\sin(x)
sin
(
x
)
False
What is the derivative of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
\sqrt{x}
x
?
1
2
x
−
1
/
2
\frac{1}{2}x^{ - 1 / 2}
2
1
x
−
1/2
Steps of the first derivative test
1️⃣ Find critical points where derivative is zero or undefined
2️⃣ Evaluate derivative signs on either side of critical points
3️⃣ Determine if critical point is a local maximum, minimum, or neither
What are the critical points of f(x) = x^{3} - 3x^{2} + 2x + 1</latex>?
x
=
x =
x
=
1
1
1
and
x
=
x =
x
=
2
2
2
Match the application with its derivative:
Position function ↔️ Velocity
Velocity function ↔️ Acceleration
Population function ↔️ Instantaneous growth rate
Cost function ↔️ Marginal cost
Revenue function ↔️ Marginal revenue
The derivative quantifies the
rate of change
at a specific point
True
What is the derivative of a constant function
f
(
x
)
=
f(x) =
f
(
x
)
=
c
c
c
?
0
The derivative of
f
(
x
)
=
f(x) =
f
(
x
)
=
a
x
a^{x}
a
x
is
a
x
ln
(
a
)
a^{x} \ln(a)
a
x
ln
(
a
)
.
True
Steps to find the derivative of
f
(
x
)
=
f(x) =
f
(
x
)
=
3
x
4
+
3x^{4} +
3
x
4
+
2
e
x
−
sin
(
x
)
2e^{x} - \sin(x)
2
e
x
−
sin
(
x
)
1️⃣ Apply the Power Rule to
3
x
4
3x^{4}
3
x
4
2️⃣ Apply the Exponential Rule to
2
e
x
2e^{x}
2
e
x
3️⃣ Apply the Trigonometric Rule to
−
sin
(
x
)
- \sin(x)
−
sin
(
x
)
4️⃣ Combine the results
The first derivative test is used to find
critical points
of a function.
True
What type of critical point occurs if the sign of the derivative changes from positive to negative?
Local maximum
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