BC Calculus AP Exam Review #2

Front

What is the logistic differential equation and what does it model?

Back

The logistic differential equation is given by \( \frac{dp}{dt} = rP\left(1 - \frac{P}{K}\right) \), where \( P \) is the population, \( r \) is the growth rate, and \( K \) is the carrying capacity. It models population growth that is initially exponential but slows as the population approaches the carrying capacity.

Front

For a logistic growth model, at what population size does the growth rate reach its maximum?

Back

The growth rate reaches its maximum at \( P = \frac{K}{2} \), where \( K \) is the carrying capacity.

Front

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

Back

The length of the curve from \( x = a \) to \( x = b \) is given by \( L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} dx \).

Front

How do you find the area between two polar curves \( r_1(\theta) \) and \( r_2(\theta) \)?

Back

The area between two polar curves is given by \( A = \frac{1}{2} \int_{\alpha}^{\beta} \left( r_1^2(\theta) - r_2^2(\theta) \right) d\theta \), where \( \alpha \) and \( \beta \) are the angles where the curves intersect.

Front

What is the limit comparison test for series convergence?

Back

The limit comparison test states that if \( a_n \) and \( b_n \) are positive sequences, and \( \lim_{n \to \infty} \frac{a_n}{b_n} = c \) where \( 0 < c < \infty \), then both series \( \sum a_n \) and \( \sum b_n \) either converge or diverge together.

Front

What is a Taylor series and how is it constructed?

Back

A Taylor series is an infinite series that represents a function as a sum of terms calculated from the values of its derivatives at a single point. The \( n \)-th degree Taylor polynomial is given by \( P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + ... + \frac{f^{(n)}(a)}{n!}(x-a)^n \).

Front

What is the significance of the first derivative of a function in relation to its graph?

Back

The first derivative of a function, \( f'(x) \), indicates the slope of the tangent line to the graph of the function at any point. It helps determine where the function is increasing or decreasing.

Front

What is the second derivative test for concavity?

Back

The second derivative test states that if \( f''(x) > 0 \), the function is concave up at that point, and if \( f''(x) < 0 \), the function is concave down.

Front

How do you determine the convergence of a series using the ratio test?

Back

The ratio test states that for a series \( \sum a_n \), if \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \): if \( L < 1 \), the series converges; if \( L > 1 \) or \( L = \infty \), the series diverges; if \( L = 1 \), the test is inconclusive.

Front

What is the formula for the area under a curve from \( a \) to \( b \)?

Back

The area under a curve from \( a \) to \( b \) is given by the definite integral \( A = \int_a^b f(x) dx \).