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Calculus I Hughes-Hallett

Calculus I Hughes-Hallett. Math 131 Br. Joel Baumeyer Christian Brothers University. Function: (Data Point of View). One quantity H, is a function of another, t, if each value of t has a unique value of H associated with it. In symbols: H = f(t).

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Calculus I Hughes-Hallett

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  1. Calculus I Hughes-Hallett Math 131 Br. Joel Baumeyer Christian Brothers University

  2. Function: (Data Point of View) • One quantity H, is a function of another, t, if each value of t has a unique value of H associated with it. In symbols: H = f(t). • We say H is the value of the function or the dependent variable or output; and • t is the argument or independent variable or input.

  3. Working Definition of Function: H = f(t) • A function is a rule (equation) which assigns to each element of the domain (independent variable) one and only one element of the range (dependent variable).

  4. Working definition of function continued: • Domain is the set of all possible values of the independent variable (t). • Range is the corresponding set of values of the dependent variable (H).

  5. Questions?

  6. General Types of Functions (Examples): • Linear: y = m(x) + b; proportion: y = kx • Polynomial: Quadratic: y =x2 ; Cubic: y= x3 ; etc • Power Functions: y = kxp • Trigonometric: y = sin x, y = Arctan x • Exponential: y = aebx ; Logarithmic: y = ln x

  7. Graph of a Function: • The graph of a function is all the points in the Cartesian plane whose coordinates make the rule (equation) of the function a true statement.

  8. Slope • m - slope : b: y-intercept • a: x-intercept • .

  9. 5 Forms of the Linear Equation • Slope-intercept: y = f(x) = b + mx • Slope-point: • Two point: • Two intercept: • General Form: Ax + By = C

  10. Exponential Functions:If a > 1, growth; a<1, decay • If r is the growth rate then a = 1 + r, and • If r is the decay rate then a = 1 - r, and

  11. Definitions and Rules of Exponentiation: • D1: • D2: • R1: • R2: • R3:

  12. Inverse Functions: • Two functions z = f(x) and z = g(x) are inverse functions if the following four statements are true: • Domain of f equals the range of g. • Range of f equals the domain of g. • f(g(x)) = x for all x in the domain of g. • g(f(y)) = y for all y in the domain of f.

  13. A logarithm is an exponent. .

  14. General Rules of Logarithms: log(a•b) = log(a) + log(b) log(a/b) = log(a) - log(b)

  15. e = 2.718281828459045... • Any exponential function can be written in terms of e by using the fact that So that

  16. Making New Functions from Old Given y = f(x): (y - b) =k f(x - a) stretches f(x) if |k| > 1 shrinks f(x) if |k| < 1 reverses y values if k is negative a moves graph right or left, a + or a - b moves graph up or down, b + or b - If f(-x) = f(x) then f is an “even” function. If f(-x) = -f(x) then f is an “odd” function.

  17. Polynomials: • A polynomial of the nth degree has n roots if complex numbers a allowed. • Zeros of the function are roots of the equation. • The graph can have at most n - 1 bends. • The leading coefficient determines the position of the graph for |x| very large.

  18. Rational Function: y = f(x) = p(x)/q(x)where p(x) and q(x) are polynomials. • Any value of x that makes q(x) = 0 is called a vertical asymptote of f(x). • If f(x) approaches a finite value a as x gets larger and larger in absolute value without stopping, then a is horizontal asymptote of f(x) and we write: • An asymptote is a “line” that a curve approaches but never reaches.

  19. Asymptote Tests y = h(x) =f(x)/g(x) • Vertical Asymptotes: Solve: g(x) = 0 If y    as x  K, where g(K) = 0, then x = K is a vertical asymptote. • Horizontal Asymptotes: If f(x)  L as x   then y = L is a vertical asymptote. Write h(x) as: , where n is the highest power of x in f(x) or g(x).

  20. Basic Trig • radian measure: q = s/r and thus s = r q, • Know triangle and circle definitions of the trig functions. • y = A sin (Bx + C) + k • A amplitude; • B - period factor; period, p = 2p/B • j - phase shift; j = -C/B • k (raise or lower graph factor)

  21. Continuity of y = f(x) • A function is said to be continuous if there are no “breaks” in its graph. • A function is continuous at a point x = a if the value of f(x)  L, a number, as x  a for values of x either greater or less than a.

  22. Intermediate Value Theorem • Suppose f is continuous on a closed interval [a,b]. If k is any number between f(a) and f(b) then there is at least one number c in [a,b] such that f(x) = k.

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