Antiderivatives are powerful tools for understanding motion and solving differential equations. They let us find position from velocity, velocity from acceleration, and calculate distances traveled. This process is like working backwards from the rate of change to the original function.
Separable differential equations take this idea further, using antiderivatives to solve complex problems. By separating variables and integrating both sides, we can find solutions to equations that describe real-world phenomena. It's like unscrambling a puzzle to reveal the underlying relationships.
Antiderivatives and Their Applications
Position from velocity function
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Antiderivatives determine position functions from given velocity functions
Velocity v ( t ) v(t) v ( t ) measures rate of change of position with respect to time
Position s ( t ) s(t) s ( t ) is the antiderivative of velocity
Integrate velocity function with respect to time to find position function
s ( t ) = ∫ v ( t ) d t s(t) = \int v(t) dt s ( t ) = ∫ v ( t ) d t
Constant of integration [ C ] ( h t t p s : / / w w w . f i v e a b l e K e y T e r m : c ) [C](https://www.fiveableKeyTerm:c) [ C ] ( h ttp s : // www . f i v e ab l eKey T er m : c ) represents initial position at time t = 0 t=0 t = 0
Example: Given v ( t ) = 3 t 2 + 2 t v(t) = 3t^2 + 2t v ( t ) = 3 t 2 + 2 t , position function is s ( t ) = ∫ ( 3 t 2 + 2 t ) d t = t 3 + t 2 + C s(t) = \int (3t^2 + 2t) dt = t^3 + t^2 + C s ( t ) = ∫ ( 3 t 2 + 2 t ) d t = t 3 + t 2 + C
If initial position is known (e.g., s ( 0 ) = 1 s(0) = 1 s ( 0 ) = 1 ), solve for C C C to determine specific position function
Velocity from acceleration function
Antiderivatives find velocity functions from given acceleration functions
Acceleration a ( t ) a(t) a ( t ) measures rate of change of velocity with respect to time
Velocity v ( t ) v(t) v ( t ) is the antiderivative of acceleration
Integrate acceleration function with respect to time to find velocity function
v ( t ) = ∫ a ( t ) d t v(t) = \int a(t) dt v ( t ) = ∫ a ( t ) d t
Constant of integration C C C represents initial velocity at time t = 0 t=0 t = 0
Example: Given a ( t ) = 6 t + 2 a(t) = 6t + 2 a ( t ) = 6 t + 2 , velocity function is v ( t ) = ∫ ( 6 t + 2 ) d t = 3 t 2 + 2 t + C v(t) = \int (6t + 2) dt = 3t^2 + 2t + C v ( t ) = ∫ ( 6 t + 2 ) d t = 3 t 2 + 2 t + C
If initial velocity is known (e.g., v ( 0 ) = 5 v(0) = 5 v ( 0 ) = 5 ), solve for C C C to determine specific velocity function
Applications of antiderivatives in motion
Antiderivatives calculate distance , displacement , and total distance traveled
Distance is total path length traveled regardless of direction
Displacement is shortest distance between start and end points
Total distance traveled sums absolute values of distances in each direction
Find distance by integrating absolute value of velocity function over time interval
Distance = ∫ t 1 t 2 ∣ v ( t ) ∣ d t \text{Distance} = \int_{t_1}^{t_2} |v(t)| dt Distance = ∫ t 1 t 2 ∣ v ( t ) ∣ d t
Example: ∫ 0 2 ∣ 3 t ∣ d t = 6 \int_0^2 |3t| dt = 6 ∫ 0 2 ∣3 t ∣ d t = 6 units of distance traveled from t = 0 t=0 t = 0 to t = 2 t=2 t = 2
Find displacement by evaluating position function at start and end times and subtracting
Displacement = s ( t 2 ) − s ( t 1 ) \text{Displacement} = s(t_2) - s(t_1) Displacement = s ( t 2 ) − s ( t 1 )
Example: If s ( t ) = t 2 + 1 s(t) = t^2 + 1 s ( t ) = t 2 + 1 , displacement from t = 1 t=1 t = 1 to t = 4 t=4 t = 4 is s ( 4 ) − s ( 1 ) = 15 s(4) - s(1) = 15 s ( 4 ) − s ( 1 ) = 15
Find total distance traveled by splitting time interval into subintervals where velocity doesn't change sign
Calculate distances in each subinterval and add together
Example: If v ( t ) = t − 2 v(t) = t-2 v ( t ) = t − 2 from t = 0 t=0 t = 0 to t = 4 t=4 t = 4 , split into [ 0 , 2 ] [0,2] [ 0 , 2 ] and [ 2 , 4 ] [2,4] [ 2 , 4 ] , find distances, and add
Solving separable differential equations
Separable differential equations are solved using antiderivatives
Separable differential equation has form d y d x = f ( x ) g ( y ) \frac{dy}{dx} = f(x)g(y) d x d y = f ( x ) g ( y )
Variables can be separated with x x x terms on one side and y y y terms on other
Steps to solve separable differential equation:
Separate variables by moving x x x terms to one side and y y y terms to other side
Integrate both sides of equation with respect to their variables
Solve for y y y as function of x x x using constant of integration C C C
Example: Solve differential equation d y d x = x y \frac{dy}{dx} = xy d x d y = x y
Separate variables: d y y = x d x \frac{dy}{y} = x dx y d y = x d x
Integrate both sides: ∫ d y y = ∫ x d x \int \frac{dy}{y} = \int x dx ∫ y d y = ∫ x d x gives ln ∣ y ∣ = 1 2 x 2 + C \ln |y| = \frac{1}{2}x^2 + C ln ∣ y ∣ = 2 1 x 2 + C
Solve for y y y : y = ± e 1 2 x 2 + C y = \pm e^{\frac{1}{2}x^2 + C} y = ± e 2 1 x 2 + C (note the ± \pm ± from absolute value)