AP Calculus AB Midterm Exam (30 Questions)

Front

What is the derivative of the function f(x) = xe^x?

Back

f'(x) = e^x + xe^x

Front

What is the integral of (2x - 3)^5 with respect to x?

Back

\int (2x - 3)^5 dx = \frac{(2x - 3)^6}{12} + C

Front

What is the formula for the trapezoidal sum for estimating the area under a curve?

Back

The trapezoidal sum is given by T = \frac{b-a}{n} \left( \frac{f(a) + f(b)}{2} + \sum_{i=1}^{n-1} f(x_i) \right) where n is the number of sub-intervals.

Front

What is the relationship between dy/dx and the function y in the equation dy/dx = 4x/y?

Back

This equation represents a separable differential equation where y is a function of x.

Front

What is the product rule for differentiation?

Back

If u(x) and v(x) are functions, then (uv)' = u'v + uv'.

Front

What is the chain rule for differentiation?

Back

If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Front

What is the Fundamental Theorem of Calculus?

Back

It states that if F is an antiderivative of f on [a, b], then \int_a^b f(x)dx = F(b) - F(a).

Front

What is the definition of a limit?

Back

A limit is the value that a function approaches as the input approaches some value.

Front

What is the definition of continuity at a point?

Back

A function f is continuous at a point c if \lim_{x \to c} f(x) = f(c).

Front

What is the second derivative test for local extrema?

Back

If f''(x) > 0 at a critical point, then f has a local minimum; if f''(x) < 0, then f has a local maximum.