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14. Practical skills and data analysis
14.2 Mathematical skills
14.2.2 Understanding calculus applications
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Cards (79)
What fundamental operation in calculus is used to find the rate of change of a function?
Differentiation
Differentiation is the inverse operation of
integration
.
True
The derivative of a position function gives the
velocity
of an object.
Differentiation can find the rate of change of temperature with respect to
time
.
Differentiation and integration are complementary tools in physics.
True
What does the derivative of a position function represent?
Velocity
What is the purpose of differentiation in determining rate of change?
Find the rate
Differentiation calculates the slope of the tangent line, which represents the instantaneous rate of change.
True
What is the first step in solving motion problems using derivatives?
Find the velocity function
If
s
(
t
)
=
s(t) =
s
(
t
)
=
3
t
2
+
3t^{2} +
3
t
2
+
2
t
−
1
2t - 1
2
t
−
1
, what is the acceleration?
6
6
6
Differentiation is the inverse operation of
integration
.
True
Differentiation finds the
rate
of change of a function.
What does integration calculate in physics?
Area under the curve
What is the derivative of velocity with respect to time called?
Acceleration
What does differentiation provide in physics in terms of a graph?
Slope of the tangent line
The derivative of a position function
s
(
t
)
s(t)
s
(
t
)
gives the velocity
v
(
t
)
v(t)
v
(
t
)
.
What does the derivative of a velocity function represent?
Acceleration
Match the function with its derivative in physics:
Position function
s
(
t
)
s(t)
s
(
t
)
↔️ Velocity
v
(
t
)
=
v(t) =
v
(
t
)
=
d
s
d
t
\frac{ds}{dt}
d
t
d
s
Velocity function
v
(
t
)
v(t)
v
(
t
)
↔️ Acceleration
a
(
t
)
=
a(t) =
a
(
t
)
=
d
v
d
t
\frac{dv}{dt}
d
t
d
v
Differentiating a position function
s
(
t
)
s(t)
s
(
t
)
results in the velocity
Steps to solve motion problems using derivatives:
1️⃣ Find the velocity function
2️⃣ Find the acceleration function
3️⃣ Evaluate the functions at specific times
Differentiation finds the rate of change, while integration calculates the accumulated
quantity
What are the applications of differentiation in physics?
Finding velocity, acceleration, rate of change
What does
d
s
d
t
\frac{ds}{dt}
d
t
d
s
represent?
Velocity
Differentiating a position function
s
(
t
)
s(t)
s
(
t
)
gives the velocity
Match the calculus operation with its result:
d
s
d
t
\frac{ds}{dt}
d
t
d
s
↔️ Velocity
d
v
d
t
\frac{dv}{dt}
d
t
d
v
↔️ Acceleration
What does the slope of the tangent line to a function represent?
Instantaneous rate of change
Integration in physics is used to calculate accumulated quantities such as displacement from
velocity
True
In calculus,
accumulation
is represented by the area under the
curve
The antiderivative of a function is the reverse of
differentiation
True
Steps to calculate displacement and velocity using integrals in motion problems
1️⃣ Calculate displacement by integrating the velocity function
2️⃣ Calculate velocity by integrating the acceleration function
To calculate displacement from velocity, one must integrate the
velocity
The formula for velocity from acceleration is
v
(
t
)
=
v(t) =
v
(
t
)
=
∫
t
1
t
2
a
(
t
)
d
t
\int_{t_{1}}^{t_{2}} a(t) dt
∫
t
1
t
2
a
(
t
)
d
t
.acceleration
Differentiation
finds the
rate of change
of a function.
What does the derivative of a position function with respect to time represent?
Velocity
The derivative of a velocity function gives the
acceleration
.
Differentiation calculates the slope of the
tangent
line, representing the instantaneous rate of change.
What does
d
v
d
t
\frac{dv}{dt}
d
t
d
v
represent?
Acceleration
What does the tangent line to a function represent?
Instantaneous rate of change
Differentiating a velocity function
v
(
t
)
v(t)
v
(
t
)
results in the acceleration
a
(
t
)
a(t)
a
(
t
)
True
Rate of change
measures how one quantity changes with respect to
another
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