MAT211 Chpt 3

Normal Line

A line that is perpendicular to the tangent line at any given point along a curve. Its slope is the negative inverse of the tangent line slope Equation for normal line y - f(x₀) = -1 / (f'(x₀)) * (x - x₀)

Derivative

A measure of how a function changes as its input changes

Rates of Change

Average: slope of corresponding secant line Instantaneous: slope of corresponding tangent line

Cost Functions

C(x): cost of manufacturing x items Can include fixed costs and variable (per item) cost C(x)/x = C(bar overhead): average cost per unit C'(x): function to describe marginal cost, the cost to produce one unit after having produced x units

Higher Order Derivatives

Derivatives of derivatives If a function is differentiable along an interval, its derivative is differentiable along that interval, as are all higher order derivatives. Each subsequent derivative measures the rate of change as the input changes for the next lower order derivative.

Slope of Tangent Line from Derivative

For f(x), m_tan = f'(a) Example f(x) = 5x² - 6x + 1 f'(x) = 10x - 6 For the point a = 2 m_tan = f'(2) = 10*2 - 6 = 14

Derivative notation

Function; derivative; 2nd order derivative... f(x); f'(x); f''(x) f(x); dy / dx (where d signifies delta) f(x); d / dx (f(x)) f(x); D_x y (D sub x y) f(x); ; ; f^(n) (x) where n is an integer denoting the order of the derivative

Implicit Differentiation

Functions where y is defined implicitly The unit circle definition is: x² + y² = 1 y is implicitly defined by its function f(x, y) = x² + y² -1 Solving for y results in two functions: y = ∓√(1-x²) Solving for y can result in more than two functions: f(x, y)=x + y³ - xy - 1 results in 3: the upper half of a parabola, the lower half, and a horizontal line

Constant Rule for Differentiation

If f(x) = c and c is a real number, f'(x) = 0 this reflects that the slope is unchanging

Cosine derivative

If f(x) = cos x then f'(x) = - sin x

Sine derivative

If f(x) = sin x then f'(x) = cos x

Power Rule for Differentiation

If f(x) = x^n and n is real number, f'(x) = nx^n-1 Examples f(x) = 2x³; f'(x) = 6x² f(x) = 2x ^ 1/2; f'(x) = 1/2 x ^ -1/2 = 1 / 2√x (recall that x^1/2 = √x and x^-2 = 1/x^2)

Quotient Rule

If f(x) and g(x) are differentiable at (x), then the derivative of f / g at x exists provided g(x)≠0 then d/dx [f(x) / g(x)] = g(x)f'(x) - f(x)g'(x) / [g(x)]²

Product Rule

If f(x) and g(x) are differentiable at x then d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

Calculate Derivative

If f(x) is a function differentiable at x then f'(x) = lim h→0 [f(x + h) - f(x)] / h To calculate the derivative of a differentiable function from the function as it approaches 0

Chain Rule

If g is a function that is differentiable at a point c and f is a function that is differentiable at g(c), then the composite function f ∘ g is differentiable at c, and the derivative is d / dx[f(g(x))] = f'(g(x)) * g'(x) or dy / dx = dy / du * du / dx, using u to express y in terms of u or (f ∘ g)'(c) = f'(g(c)) * g'(c)

Chain Rule for Powers

If g is differentiable for all x in its domain and n is an integer (g'(x))^n = n(g(x))^n-1 * g'(x)

Growth Models

Models that describe the growth of something as time changes. Can be used to describe population, prices, etc.

One Dimensional Kinematics

Position function s=f(t) s: origin of object f(t): function that describes its position at a given time f'(t): velocity f''(t): acceleration

Power Rule for Rational Exponents

Power Rule applies to exponents that are integers or rational numbers. It is used the same way in either case. f(x)=x^n/m; f'(x)=n/m * x ^ (n/m) - 1 f(x)=√x = x^1/2; f'(x) = 1 / 2 * x ^ -1/2 = 1 / 2x^1/2 = 1 / 2√x

Differentiable

The function must continuous at n have only non-vertical tangent lines at n must be relatively smooth (no cusps or corners) Example: f(x) = |x| is not differentiable because a corner exists at 0

Continuous

The function must be defined at n and lim x→n⁺ = lim x→n⁻

Graph of Derivative

The graph of a derivative reflects the function in that it describes the rate of change of that function Heavy line is derivative of function (parabola). The y value corresponds with the slope of m_tan at that x value. When the slope of f(x) = 0, f'(x) is at y = 0 f'(x) crosses the x axis when m_tan f(x) = 0 When the slope of f(x) is positive, f'(x) is at y > 0 f'(x) > 0 ⇒ f(x) is increasing when the slope of f(x) is negative, f'(x) is at y < 0 f'(x) < 0 ⇒ f(x) is decreasing

Combining Rules for Differentiation

These rules can be combined: f(x) = 4x³+9x²-72x-2 Find the derivative for the first three terms using the power rule. Find the derivative for the last term using the constant rule. Combine all using the sum and difference rules: f'(x) = 12x²+18x-72-0 (no need to note -0)

Sum Rule for Differentiation

d / dx [f(x)+g(x)] = f'(x)+g'(x) Assuming that f and g are differentiable at x Useful for finding derivatives for polynomials

Difference Rule for Differentiation

d / dx [f(x)-g(x)] = f'(x)-g'(x) Assuming that f and g are differentiable at x

Chain Rule for 3 part composite

d / dx [sin(cos x²)] of the pattern f(g(h(x))) sin: outer to inner cos x² cos x²: outer to inner x² Chain rule for sin: cos(cos x²) * d / dx (cos x²) Chain rule for cos x²: cos(cos x²) * -sin x² * d / dx (x²) differentiate x²: cos(cos x²) * -sin x² * 2x Simplify: -2x cos (cos x²) sin x²

Constant Multiple Rule for Differentiation

d / dx c*f(x); c'f(x) provided that f is differentiable at x and c is a constant

Demand & Revenue Functions

d(p): demand (in units) per price R(p) = pd(p): Revenue R'(p): revenue derivative, revenue is maxed where R'(p)=0

Average Rate

f(a + ∆t) - f(a) / ∆t Applies to finding the average for any function regardless of what the function describes

Trigonometric Derivatives

f(x) = sin x; f'(x) = cos x f(x) = csc x; f'(x) = -csc x cot x f(x) = cos x; f'(x) = -sin x f(x) = sec x; f'(x) = sec x tan x f(x) = tan x; f'(x) = sec² x f(x) = cot x; f'(x) = -csc² x

Trigonometric Limits

lim x→0 sin x / x = 1 lim x→0 cos x - 1 / x = 0

Slope of Tangent Line

m_tan = lim x→a [f(x) - f(a)] / (x - a) Calculate slope of tangent from function and a single point a on that function.

Profit functions

p(x): per unit sale price for x units P(x) = xp(x)-C(x): profit for producing and selling x units P(x)/x: average profit per item dP/dx = P'(x): marginal profit

Explicit Differentiation

possible for functions where y is defined explicitly as a function of x

Related Rates

problems in which multiple variables, related in a known way, change in their result to a third variable e.g. How large and how fast does a leak spread over time?

Tangent Lines & Implicit functions

1. Evaluate whether the point is part of the function: An (x, y) point lies on the curve of a given implicit function if x and y satisfy the function. 3x²+xy+2y²=24; (1,3) and (1, -3) lie on this curve; (1, 1) and (3,3) do not 2. Differentiate the function 3. Substitute the point's values into the derivative, solve to obtain slope of tangent line. 4. Sub slope into line formula y - f(a) = m_tan(x - a).

Steps for Related Rate Problems

1. Sketch, id the given rates and the rate(s) to determine 2. Write equation(s) to express the relationship(s) between variable(s) 3. Introduce rates of change by differentiating the appropriate equation(s) (here, typically with respect to t) 4. Sub known values, solve 5. Double check units and reasonableness of answer

Velocity

change in position / change in time First derivative of a 1d kinematic function

Acceleration

change in velocity / change in time Second derivative of a 1d kinematic function

Equation of Tangent Line

y - f(a) = m_tan(x - a)