AP Calculus AB Final Review

Front

What is the average value of a function f(x) over an interval [a, b]?

Back

The average value of a function f(x) over the interval [a, b] is given by the formula: \( \text{Average Value} = \frac{1}{b-a} \int_a^b f(x) \, dx \).

Front

What is the trapezoidal sum for approximating the area under a curve?

Back

The trapezoidal sum is an approximation of the integral of a function, calculated by dividing the area under the curve into trapezoids and summing their areas. The formula is: \( T_n = \frac{b-a}{2n} \left( f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right) \).

Front

How do you find the rate of change of surface area of a cube as its edge length decreases?

Back

To find the rate of change of surface area (dA/dt) of a cube, use the formula for surface area \( A = 6s^2 \) and apply the chain rule: \( \frac{dA}{dt} = 12s \frac{ds}{dt} \).

Front

What is the relationship between volume and surface area for a cube?

Back

For a cube with edge length s, the volume is given by \( V = s^3 \) and the surface area is \( A = 6s^2 \). As the edge length changes, both volume and surface area change accordingly.

Front

What is an Initial Value Problem (IVP) in calculus?

Back

An Initial Value Problem is a differential equation along with a specified value at a given point, typically expressed as \( y(t_0) = y_0 \). It requires finding a function that satisfies both the differential equation and the initial condition.

Front

What is the Fundamental Theorem of Calculus?

Back

The Fundamental Theorem of Calculus links differentiation and integration, stating that if \( f \) is continuous on [a, b], then: 1) \( F(x) = \int_a^x f(t) \, dt \) is continuous on [a, b] and differentiable on (a, b), and 2) \( F'(x) = f(x) \).

Front

What is the derivative of a function at a point?

Back

The derivative of a function f at a point x is defined as the limit: \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \). It represents the slope of the tangent line to the graph of f at that point.

Front

What is the concept of limits in calculus?

Back

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It is denoted as \( \lim_{x \to c} f(x) = L \), meaning f(x) approaches L as x approaches c.

Front

What is the power rule for differentiation?

Back

The power rule states that if \( f(x) = x^n \), then the derivative is given by \( f'(x) = n \cdot x^{n-1} \). This rule applies to any real number n.

Front

What is the chain rule in differentiation?

Back

The chain rule is a formula for computing the derivative of a composite function. If \( y = f(g(x)) \), then the derivative is given by \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).