Podcast
Questions and Answers
The rate of change of z with respect to x is represented by ______
The rate of change of z with respect to x is represented by ______
Δz/Δx
The total differential of z = f(x, y) is given by ______
The total differential of z = f(x, y) is given by ______
dz = ∂f/∂x dx + ∂f/∂y dy
The ______ gives an approximation of the change in the dependent variable given small changes in each of the independent variables.
The ______ gives an approximation of the change in the dependent variable given small changes in each of the independent variables.
total differential
The total derivative represents the total change in the dependent variable due to changes in all the ______ variables.
The total derivative represents the total change in the dependent variable due to changes in all the ______ variables.
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The effect on z of a small change in x is given by the ______ differential.
The effect on z of a small change in x is given by the ______ differential.
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The total differential is the sum of the ______ differentials.
The total differential is the sum of the ______ differentials.
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The total derivative of z with respect to x is ∂z/∂x = ∂f/∂x + ∂f/∂y ⋅ ______ dy/dx
The total derivative of z with respect to x is ∂z/∂x = ∂f/∂x + ∂f/∂y ⋅ ______ dy/dx
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∂f/∂x = ______ x^2
∂f/∂x = ______ x^2
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The total derivative of z with respect to t becomes ∂z/dt = ∂f/∂x ⋅ ______ dt + ∂f/∂y ⋅ dy/dt
The total derivative of z with respect to t becomes ∂z/dt = ∂f/∂x ⋅ ______ dt + ∂f/∂y ⋅ dy/dt
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The direct effect of x on z, plus the indirect effect of x on z through y is given by the ______ derivative.
The direct effect of x on z, plus the indirect effect of x on z through y is given by the ______ derivative.
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∂f/∂x = ______ x
∂f/∂x = ______ x
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The ______ refers to the rate of change of a function.
The ______ refers to the rate of change of a function.
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A supply function is given by the equation P = ______ Q2
A supply function is given by the equation P = ______ Q2
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Find the derivative of P, the differential of P, and the approximate change in P if Q is increased by ______ %
Find the derivative of P, the differential of P, and the approximate change in P if Q is increased by ______ %
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The total differential of z is given by dz = ______dx + ______dy.
The total differential of z is given by dz = ______dx + ______dy.
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The ______ of z is given by the partial derivative of z with respect to x and y.
The ______ of z is given by the partial derivative of z with respect to x and y.
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The formula for the total derivative is given by dz/dx = ______ + ______ * dy/dx.
The formula for the total derivative is given by dz/dx = ______ + ______ * dy/dx.
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The ______ refers to the actual change in the function that occurs when the independent variable is altered.
The ______ refers to the actual change in the function that occurs when the independent variable is altered.
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The derivative of P is ______________ = 6Q
The derivative of P is ______________ = 6Q
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The differential of P is dP = ( ______________ )dQ
The differential of P is dP = ( ______________ )dQ
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The change in Q is ΔQ = ______________ % · Q
The change in Q is ΔQ = ______________ % · Q
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The approximate change in P is ΔP = ( ______________ )ΔQ
The approximate change in P is ΔP = ( ______________ )ΔQ
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The annual revenues (R) of Highple Consulting depends on the fees (F), promotions (P), and the number of marketers ( ______________ ).
The annual revenues (R) of Highple Consulting depends on the fees (F), promotions (P), and the number of marketers ( ______________ ).
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The supply function is given by the equation P = ______________ Q2.
The supply function is given by the equation P = ______________ Q2.
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Study Notes
Rate of Change
- The rate of change of z with respect to x is represented by ∂z/∂x.
- The total differential of z = f(x, y) is given by dz = (∂f/∂x)dx + (∂f/∂y)dy.
- The total differential gives an approximation of the change in the dependent variable given small changes in each of the independent variables.
- The total derivative represents the total change in the dependent variable due to changes in all the independent variables.
- The effect on z of a small change in x is given by the partial differential.
- The total differential is the sum of the partial differentials.
- The total derivative of z with respect to x is ∂z/∂x = ∂f/∂x + ∂f/∂y ⋅ dy/dx.
Partial Derivatives
- ∂f/∂x = 2x.
Total Derivative
- The total derivative of z with respect to t becomes ∂z/dt = ∂f/∂x ⋅ dx/dt + ∂f/∂y ⋅ dy/dt.
- The direct effect of x on z, plus the indirect effect of x on z through y is given by the total derivative.
- ∂f/∂x = x.
Derivative, Differential and Change
- The derivative refers to the rate of change of a function.
- A supply function is given by the equation P = aQ^2.
- Find the derivative of P, the differential of P, and the approximate change in P if Q is increased by 5%.
Total Differential, Partial Derivative, and Total Derivative
- The total differential of z is given by dz = (∂z/∂x)dx + (∂z/∂y)dy.
- The derivative of z is given by the partial derivative of z with respect to x and y.
- The formula for the total derivative is given by dz/dx = (∂z/∂x) + (∂z/∂y) * dy/dx.
- The differential refers to the actual change in the function that occurs when the independent variable is altered.
Example: Supply Function
- The derivative of P is dP/dQ = 6Q.
- The differential of P is dP = (6Q)dQ.
- The change in Q is ΔQ = 5% · Q.
- The approximate change in P is ΔP = (6Q)ΔQ.
Revenue Function
- The annual revenues (R) of Highple Consulting depends on the fees (F), promotions (P), and the number of marketers (M).
- The supply function is given by the equation P = aQ^2.
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Description
Learn about incremental changes, partial differentials, and total derivatives. Understand how small changes in independent variables affect dependent variables and calculate rates of change.
