Ch. 6/7 Review Calculus

Front

What is a differential equation?

Back

A differential equation is an equation that relates a function with its derivatives. It describes how a quantity changes in relation to another variable.

Front

Solve the differential equation: dy/dt = 3t^2/y with initial condition y(2) = 0.

Back

y = (2t^3 - 16)^{1/2}

Front

What is the general form of a logistic equation?

Back

The general form of a logistic equation is P(t) = K / (1 + Ae^{-rt}), where K is the carrying capacity, A is a constant, r is the growth rate, and t is time.

Front

Find a solution for y if dy/dx = 2x√y and y = 4 when x = 3.

Back

y = ¼(x^2 - 5)^{2}

Front

What is the carrying capacity in a logistic growth model?

Back

The carrying capacity is the maximum population size that an environment can sustain indefinitely.

Front

Write the integral that can be used to find the area between the curves x = -3y² + 4 and x = y³.

Back

Front

What is the formula for the length of a curve described by a function y = f(x)?

Back

Front

Find the length of the curve described by y = \frac{2}{3}x^{\frac{3}{2}} for 0 ≤ x ≤ 8.

Back

Length = \frac{52}{3}.

Front

What is the method of separation of variables in solving differential equations?

Back

The method of separation of variables involves rearranging a differential equation to isolate the variables on opposite sides of the equation.

Front

What is the significance of initial conditions in differential equations?

Back

Initial conditions provide specific values for the function and its derivatives at a given point, allowing for the determination of a unique solution.