Cal rules final

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• Theorem 1.5.5 states that if limx->c f(x) = L and if the function f is continuous at L, then the limit symbol can be moved through a function sign.

  • The function sign provided the limit of the expression inside the function sign exists and the function is continuous at this limit.

  • This equality remains valid if the limit is replaced by lim c+, lim c-, lim +oo, or lim -oo+.

• In the special case where f(x) = |x|, if limx->c f(x) exists, then lim |g(x)| = lim g(x).

• Theorem 1.5.6 states that if the function g is continuous at c and the function f is continuous at g(c), then the composition fog is continuous at c.

• If the function g is continuous everywhere and the function f is continuous everywhere, then the composition fog is continuous everywhere.

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• This section discusses the continuity properties of trigonometric functions.

• The graphs of sin x and cos x are drawn as continuous curves.

• The angle x approaches the angle c, and the point P (cos x, sin x) moves along the unit circle toward Q (cos c, sin c).

• The coordinates of P approach the corresponding coordinates of Q, implying that sin x and cos x are continuous at any point C.

• The formulas in (1) can be used to find limits of other trigonometric functions by expressing them in terms of sin x and cos x.

• The six basic trigonometric functions (sin x, cos x, tan x, sec x, csc x, cot x) are continuous on their domains.

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• Theorem 1.6.1 states that if C is any number in the natural domain of a trigonometric function, then the function is continuous at C.

• The limit of cos x as x approaches C is cos C.

• Example 1: Find the limit of cos x.

  • Since the cosine function is continuous everywhere, the limit of cos x is cos C if lim g(x) exists.

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• The limit x lim sin x = 1 is not easy to establish with certainty.

• The method of squeezing is used to prove limits of indeterminate forms.

• The Squeezing Theorem states that if g and h have the same limit as x approaches c, then f also has this limit as x approaches c.

• Theorems 1.6.3 (a) and (b) state the limits of sin x and 1 - cos x.

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• Definition of the tangent line to a curve y = f(x) at a point P(x0, f(x0)).

• The tangent line has the equation y - f(x0) = m(x - x0), where m is the slope of the tangent line.

• The derivative function is defined as the limit of the difference quotient.

• The derivative function represents the slope of the tangent line or the instantaneous rate of change of y with respect to x.

• Example 1: Find the derivative of f(x) = x^2 and use it to find the equation of the tangent line at x = 2.

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• The concept of a "derivative" is discussed as the primary mathematical tool for calculating and studying rates of change.

• The derivative of a function f with respect to x is defined as the limit of the difference quotient.

• The domain of the derivative function consists of all x for which the limit exists.

• Example 1: Find the derivative of f(x) = x^2 and use it to find the equation of the tangent line at x = 2.

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• Differentiability is defined as the existence of the limit that defines the derivative of a function.

• A function f is said to be differentiable at x0 if the limit exists.

• If a function is differentiable at each point of an open interval, it is differentiable on that interval.

• If a function is differentiable at each point of a specific type of open interval, it is differentiable everywhere.

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• Differentiability of a function at a point XO is determined by the existence of a tangent line at XO on the graph of the function.

• A function is not differentiable at a point XO if the secant lines from P(XO, f(XO)) to points Q(X, f(X)) distinct from P do not approach a unique nonvertical limiting position as X approaches XO.

• Two common ways in which a function that is continuous at XO can fail to be differentiable at XO are corner points and points of vertical tangency.

  • At a corner point, the slopes of the secant lines have different limits from the left and from the right, and the two-sided limit that defines the derivative does not exist.

  • At a point of vertical tangency, the slopes of the secant lines approach +00 or -00 from the left and from the right, and the limit that defines the derivative does not exist.

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• The relationship between differentiability and continuity is that if a function is differentiable at a point, then it must be continuous at that point.

• Theorem: If a function f is differentiable at XO, then f is continuous at XO.

• Proof: Given that f is differentiable at XO, it follows that f'(XO) exists and is given by the limit (7).

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• To show that f is continuous at XO, we must show that the limit of f(x) as X approaches XO is equal to f(XO).

• This can be expressed in terms of the variable h (1=X- XO) as the limit of [f(x+h) - f(x)] as h approaches 0 is equal to 0.

• Using the given limit (7), it can be proved that the limit is indeed 0.

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• The derivative of a constant function is 0.

• The simplest kind of function is a constant function f(x) = c.

• The graph of a constant function is a horizontal line with slope 0, so the tangent line to the graph also has slope 0.

• Algebraically, the derivative of a constant function can be calculated using the limit formula.

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• The derivative of power functions can be determined using the power rule.

• The power rule states that if n is a positive integer, then the derivative of x^n is n*x^(n-1).

• Examples of power functions and their derivatives are provided.

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• The derivative of a constant times a function can be calculated using the constant multiple rule.

• The constant multiple rule states that if f is differentiable at x and C is any real number, then the derivative of C*f is C times the derivative of f.

• The proof of the constant multiple rule involves applying the limit definition of the derivative.

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• A constant factor can be moved through a derivative sign.

• The derivative of a sum or difference of functions can be calculated using the sum and difference rules.

• The sum rule states that if f and g are differentiable at x, then the derivative of f + g is the sum of the derivatives of f and g.

• The difference rule states that if f and g are differentiable at x, then the derivative of f - g is the difference of the derivatives of f and g.

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• The product rule allows us to find the derivative of the product of two functions.

• If f and g are differentiable at x, then the derivative of f*g is given by the product of the derivative of f and g, plus the product of f and the derivative of g.

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• The derivative of a quotient of functions can be calculated using the quotient rule.

• If f and g are both differentiable at x and g(x) is not equal to 0, then the derivative of f/g is given by the numerator times the derivative of the denominator, minus the denominator times the derivative of the numerator, all divided by the square of the denominator.

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• Summary of differentiation rules:

  • The derivative of a constant is 0.

  • The derivative of a sum or difference of functions is the sum or difference of their derivatives.

  • The derivative of a constant times a function is the constant times the derivative of the function.

  • The derivative of a product of functions is given by the product of the derivative of one function and the other function, plus the product of the first function and the derivative of the second function.

  • The derivative of a quotient of functions is given by the numerator times the derivative of the denominator, minus the denominator times the derivative of the numerator, all divided by the square of the denominator.

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• Objective: Obtain formulas for the derivatives of the six basic trigonometric functions

• Assumptions: Variable x is measured in radians, limits in Theorem 1.6.3 restated using h as the variable

• Derivative of f(x) = sinx: Using the definition of the derivative, we obtain h cos h + cosx - sin x lim

• Algebraic reorganization: lim COSXXCOS.X cos x does not involve the variable h and hence h-> is treated as a constant in the limit computation

• Validity of formulas: Formulas (1) and (2) and the derivation of Formulas (3) and (4) are only valid if h and x are in radians

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• Derivatives of the remaining trigonometric functions:

  • d [tan x ] = sec2 x

  • d [sec x] = sec x tan x

  • d [cot x] = - csc2 x

  • d [csc x] = - CSC x cot x

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• Objective: Derive a formula for the derivative of a composition of functions

• Example: Miles traveled in a car as a function of gallons of gas and cost of gas

• Composition of functions: y = f(u), u = g(x)

• Rates of change: f'(u) = dy = 20 miles per gallon, each dollar spent gives 1/4 of a gallon

• Miles traveled per dollar: dy/dx = -20/1 * 1/4 = -5 miles per dollar

• Chain Rule Theorem: If g is differentiable at x and f is differentiable at g(x), then the composition fog is differentiable at x

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• Alternative version of the Chain Rule: [f(g(x))] = (fog)'(x) = f(g(x))g'(x)

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• Generalized derivative formulas: dx = (3)

• Examples of generalized derivative formulas:

  • die [cos x] = -sin u du

  • die [tan u] = sec2 u du

  • die [cot x] = -csc2 u du

  • die [sec u] = -csc u cot u du

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• Theorem: Derivative conditions for increasing, decreasing, and constant functions

  • If f'(x) > 0 for every value of x in (a, b), then f is increasing on [a, b]

  • If f'(x) < 0 for every value of x in (a, b), then f is decreasing on [a, b]

  • If f'(x) = 0 for every value of x in (a, b), then f is constant on [a, b]

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• Relative maxima and minima: Tops of hills are relative maxima, bottoms of valleys are relative minima

• Definition: Relative maximum at XO if f(xo) is the largest value in an open interval containing XO

• Definition: Relative minimum at XO if f(xo) is the smallest value in an open interval containing XO

• Relative extremum: If f has either a relative maximum or a relative minimum at XO, it has a