Calculus Limits and Derivatives Quiz

Study Notes

Limits

Limits describe how a function behaves as its input approaches a specific value.
A limit can exist even if the function is undefined at that point.
𝝏 ➡️ a means that as x gets closer to a, the function values get arbitrarily close to L.
One-sided limits (left-hand and right-hand limits) help determine overall limits.
A limit exists if both one-sided limits are equal.
Limits at infinity analyze function behavior as x approaches positive or negative infinity.

Derivatives

Derivatives measure the rate of change of a function.
Defined as the limit of the difference quotient: f'(x) = lim (h→0) [f(x+h) - f(x)] / h
Geometrically, the derivative represents the slope of the tangent line to the function at a point.
Physically, the derivative of position is velocity, and the derivative of velocity is acceleration.

Tangent Lines and Velocity Problems

A tangent line touches a curve at a single point without crossing it.
The equation of a tangent line with slope m at point (x₁, y₁) is y - y₁ = m(x - x₁).
Velocity is the derivative of position.
Acceleration is the derivative of velocity.

Rates of Change

Derivatives represent rates of change.
The rate of change of a function f(x) is given by f'(x).

Rules of Differentiation

Power Rule: d/dx (xⁿ) = nxⁿ⁻¹
Product Rule: d/dx (uv) = u'v + uv'
Quotient Rule: d/dx (u/v) = (v u' - u v') / v²
Chain Rule: d/dx (f(g(x))) = f'(g(x)) * g'(x)

Extrema

Critical points are where the derivative is zero or undefined.
A local minimum occurs where the derivative changes from negative to positive.
A local maximum occurs where the derivative changes from positive to negative.

Theorems

Rolle's Theorem: If a function is continuous on a closed interval and differentiable on the open interval, and the function values at the endpoints are equal, then there exists a point where the derivative is zero.
Mean Value Theorem: If a function is continuous on a closed interval and differentiable on the open interval, then there exists a point where the instantaneous rate of change is equal to the average rate of change over the interval.

Optimization

Optimization techniques find maximum or minimum values of a function.
Steps typically involve finding critical points, analyzing the second derivative, or evaluating the behavior of the function.

Integrals (Areas and Distances)

Integrals calculate the accumulated quantity of a function over an interval.
The definite integral calculates the area under a curve between two points.
The Fundamental Theorem of Calculus relates differentiation and integration: ∫f(x)dx = F(x) + C, where F(x) is the antiderivative of f(x).
If a function represents velocity, the definite integral represents the distance travelled.

Description

Test your understanding of limits and derivatives in calculus. This quiz covers key concepts such as one-sided limits, the definition of derivatives, and applications to tangent lines and velocity problems. Perfect for students looking to reinforce their knowledge in this fundamental area of mathematics.