Optional Honors Calculus Midterm Exam Review

Front

What does it mean for a function to be continuous at a point c?

Back

A function is continuous at a point c if and only if f(c) = lim f(x) as x approaches c.

Front

How do you find the derivative of a function at a specific point?

Back

To find dy/dx at a given point, you calculate the derivative of the function and then substitute the x-value of the point into the derivative.

Front

What is the derivative of y = e^(5x) + 1?

Back

The derivative y' = 5e^(5x).

Front

How do you find the derivative of a quotient of two functions g(x) = (3x - 2)/(x^2 + 2)?

Back

Use the quotient rule: g'(x) = [ (x^2 + 2)(3) - (3x - 2)(2x) ] / (x^2 + 2)^2.

Front

What is the derivative of f(x) = 5/x^2?

Back

The derivative f'(x) = -10/x^3.

Front

Define the limit of a function as x approaches a number c.

Back

The limit of a function as x approaches c is the value that f(x) gets closer to as x gets closer to c.

Front

What is the power rule for differentiation?

Back

The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1).

Front

What is the product rule for differentiation?

Back

The product rule states that if u(x) and v(x) are functions, then (uv)' = u'v + uv'.

Front

What is the chain rule for differentiation?

Back

The chain rule states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Front

What is the significance of the first derivative of a function?

Back

The first derivative indicates the rate of change of the function and can be used to find critical points and determine increasing or decreasing behavior.