MTH 114 Study Guide - Final Guide: Raw Image Format, Power Rule, Eval

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MTH 114 Lecture 67
if particle in rectilinear motion moves along x-axi so that pos coord frame of elapsed
time t: s = f(t)
- f called pos func of particle
- avg velocity over time interval [t_0, t_0+h], h > 0: v_avg = delta pos/ delta t = f(x_0+h) -
f(x_0)/h
instantaneous velocity
-limit as h approaches 0 of its avg veto or v_avg over time intervals between t = t_0 & t
= t_0 + h
- v avt = lim of f(x_0+h)-f(x_0)/h
def 2.2.1
- function f prime defined by formula f'(x) = limit of f(x+h)-f(x)/h as h approaches 0
- called derivative of f with respect to x
- domain of f prime: all x in domain of f for which limit exists
finding eq for tan line to y = f(x) at x = x_0
1. evaluate f(x_0): point of tangency is (x_0, f(x_0))
2. find f'(x) and eval f'(x_0) which is slope m of line
3. substitute value of slope m and point (x_0, f(x_0)) into point-slope form of line:
- y - f(x_0) = f'(x_0)(x-x_0)
- y = f(x_0) +f'()
2.2.2 def
func f said to be differentiable at x_0 if limit f'(x_0) if lim of f(x_0+)
Properties of differentiability
- not differentiable at any point x_0 where sec line from P(x_0, f(x_0)) to points Q(x, f(x))
distinct from P do not approach unique non vertical limiting pos as x approaches x_0
- corners/one-sided limit does not exist
- points of vertical tan
2.2.3 theorem
if func f differentiable at x_0: f is cont at x_0 since f'(x) exists and, therefore, the limit
exists as h approaches 0 & functions not differentiable at points of discont
if function f defined on closed interval [a,b] but not outside interval
f not defined at endpoints of interval because derive are 2-sided limits
increment
change from initial value and h is the increment
2.3.1 theorem
deriv of oonstant function: 0
2.3.2 theorem (power rule)
if n is pos number and any real number: derivative of x^n = nx^n-1
2.3.4 theorem (constant multiplication)
if x is differentiable at x and c is any real number: c(f) is also diff at x and the derivative
of c(f(x)) is equal to c(derivative of f(x))
2.3.5 theorem (sum and diff rules)
if f & g are diff at x: f+g and f-g are also diff
higher order derivatives
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Document Summary

Mth 114 lecture 67 if particle in rectilinear motion moves along x-axi so that pos coord frame of elapsed time t: s = f(t) Avg velocity over time interval [t_0, t_0+h], h > 0: v_avg = delta pos/ delta t = f(x_0+h) - f(x_0)/h instantaneous velocity. Limit as h approaches 0 of its avg veto or v_avg over time intervals between t = t_0 & t. V avt = lim of f(x_0+h)-f(x_0)/h def 2. 2. 1. Function f prime defined by formula f"(x) = limit of f(x+h)-f(x)/h as h approaches 0. Called derivative of f with respect to x. 2. 2. 2 def func f said to be differentiable at x_0 if limit f"(x_0) if lim of f(x_0+) Not differentiable at any point x_0 where sec line from p(x_0, f(x_0)) to points q(x, f(x)) distinct from p do not approach unique non vertical limiting pos as x approaches x_0.