Unit 4 Calculus

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero.

The equation of the tangent line at a point (a, f(a)) is given by: \( y = f(a) + f'(a)(x - a) \)

To approximate f(x) using the tangent line at x = a, use the formula: \( f(x) \approx f(a) + f'(a)(x - a) \)

The area A of an equilateral triangle with side length s is given by: \( A = \frac{s^2\sqrt{3}}{4} \)

Use the chain rule: \( \frac{dA}{dt} = \frac{dA}{ds} \cdot \frac{ds}{dt} \) where \( \frac{dA}{ds} = \frac{s\sqrt{3}}{2} \)

In a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): \( c^2 = a^2 + b^2 \)

If two objects are moving at rates v1 and v2 at angles θ1 and θ2 respectively, the distance D between them can be found using: \( D = \sqrt{(v_1 t \cos(\theta_1))^2 + (v_2 t \sin(\theta_2))^2} \)

In related rates problems, the rates of change of the variables are connected through a relationship that involves both variables, often expressed as an equation.

The chain rule states that if a variable y depends on u, and u depends on x, then the derivative of y with respect to x is: \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \)

The second derivative of a function provides information about the concavity of the function and can indicate points of inflection where the function changes from concave up to concave down or vice versa.