Calculus

1. A1 A4 A3 A4 A5 A10 … … A2 A3 A = A1 + A2 +A3 +A4 A = limAn = πr2 n -> ∞ y y y y=x2 A x x x 0 0 0 Calculus • Area Problem Volume Problem Integral Calculus

2. y y y=x2 y=x2 Q secant line tangent line P P kPQ k x x 0 0 k = limkPQ QP Integral calculus ↔ Differential calculus inverse problems Calculus • Tangent Problem Tangent to a curve at a point P. Slope? Differential Calculus Calculus: deals with limits

3. Example: A(r)=πr2 r x1 f(x1) x2 f(x2) rule, machine input output x3 x black box f(x) f(x3) function y y=x2 1 x 0 1 FUNCTIONS Main objects in Calculus: A functionf is a rule that assigns to each element x in a set A (domain of f) exactly one element f(x) in a set B (rangeof f). x – independent variable, f(x) – dependent variable. domain A range B = all possible values Graph of a function: { (x,f(x)) | xA}

4. y y=x2 y=1-x 1 x 0 1 y=x2 y increasing on [0,∞) f(x2) f(x1) decreasing on (∞,0] x 0 x1 x2 Some properties: • Piecewise defined • Symmetric • Increasing/Decreasing • Periodic even:f(-x) = f(x), e.g. f(x) = x2, symmetric w.r.t.y-axis odd:f(-x) = - f(x), e.g. f(x) = x3, symmetric about the origin decreasing on I: if f(x1)>f(x2) for any x1<x2 in I increasing on I: if f(x1)<f(x2) for any x1<x2 in I with period T, if f(x+T) = f(x), e.g. cos(x+2π) = cos(x)

5. Power:f(x) = xa, a – constant. n > 0 integer if a = n polynomial (i.e. ); if a = 1/n  root (i.e. ); if a = -n  reciprocal (i.e. ) y y=1/x hyperbola Rational:f(x) = P(x)/Q(x) – ratio of two polynomials, domain: such x that Q(x) ≠ 0 (e.g. , domain x ≠  2) 0 x Algebraic: algebraic operations on polynomials (i.e. + , − ,  ,  , ) (e.g. ) Some basic functions: Linear:f(x) = kx+b, graph is a line with slopek and y-intersectb, grow at constant rate Polynomial:f(x) = anxn+an-1xn-1+…+a1x+a0 , n ≥0 integer, coefficients ai – constants, i=0..n if an ≠ 0 then n is called degree of polynomial, domain (−∞,∞)

6. y y 1 y=cos(x) 1 y=sin(x) x x 0 0 -π/2 π -π/2 π π/2 π/2 y y=tan(x) 1 x 0 -π/2 π π/2 Trigonometric:f(x) = {sin(x), cos(x), tan(x), cot(x), sec(x), csc(x)}, x in radians domain: (−∞,∞) range: [-1,1] period: 2π(waves) zeros: πn for sin(x) π/2+πn for cos(x) domain:cos(x) ≠ 0 range: (−∞,∞) period:π zeros: πn for sin(x)

7. Exponential:f(x) = ax, constant a > 0 – base, x - exponent Special basee = 2.71828… Logarithmic:f(x) =logax. Inverse of exponential: logax = y ay = x. lnx:= loge`x, ln e=1 Hyperbolic: certain combination of ex and e-x:

8. y y y=coshx 1/2 1 y=sinhx y=ex/2 1/2 0 y=ex/2 y=e-x/2 x y=-e-x/2 0 1/2 x y asymptote y=1 1 0 x y=tanhx asymptote y=-1 1 -1 domain: (−∞,∞) range: (−∞,∞) for sinh(x) [1,∞)for cosh(x) Application of cosh: shape of hanging wire = catenary (catena=chain in Latin) y=c+acosh(x/a)

9. circle (cosa,sina) (cosha,sinha) r=1 hyperbola 0 0 y=tanh-1x y=sinh-1x 0 0 -1 1 0 1 y=cosh-1x Trigonometric vs. Hyperbolic a= twice the area of this region a= twice the area of this region Inverse hyperbolic:

10. Show that Proof. where Then Solving this quadratic equation with respect to z: But while z should be positive. Therefore

11. Limit of a function: • A function f(x) is continuous at a pointa if i.e. • f(a) defined • exists • and f(b) N f(a) a b c A function f(x) is continuous on an interval if it is continuous at every point of this interval. Itermediate Value Theorem:f – continuous on [a,b]. i.e. continuous function takes on every intermediate value between the function values f(a) and f(b).

12. y=f(x) Q(a+h,f(a+h)) f(a+h)-f(a) P(a,f(a)) q q h Tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope if this limit exists. The tangent line to the curve y=f(x) at the point (a,f(a)) is the line through (a,f(a)) whose slope is equal to f(a) = the derivative of f at a, i.e. If this limit exists then the function f is differentiable at a pointa. Example. Tangent line has the same direction as the curve at the point of contact. (tangent = touching in Latin)

13. Example. Other notations: Th. If f is differentiable at a, then it is continuous at a. (Proof: see section 2.9) a a a Example: f(x)=|x|, a=0 not differentiable at a, but continuous Example: discontinuity not continuous at a  not differentiable at a Example: vertical tangent f is continuous at a and not differentiable: Derivative as a function (let the point a vary): Given any x for which this limit exists  assign to x the number f(x).

14. Constant Multiple Rule: Sum Rule: Difference Rule: e is defined s.t. Product Rule: Quotient Rule: Derivatives of some basic functions. Rules. Constant c: Power: Polynomial: derivative of polynomial = sum of all corresponding derivatives: Exponential:  rate of change of any exponential function is proportional to the function itself. Trigonometric:

15. Chain Rule: Implicit differentiation… for inverse functions Hyperbolic: Logarithmic: Inverse hyperbolic:

16. Show that Proof. Let us differentiate both sides with respect to x and find y’: But since Therefore,