Math Test 1 Flashcards
Sketch x^2, x^3, logx, e^x, sinx, cosx, tanx, 1/x, sqrt(x), absolute x
Which term in a polynomial function will dominate for small x & large x
For small x, smaller degrees of power will dominate.
For large x, larger degrees of power will dominate
Sketch & asymptotically show e^x dominates any power function
(e^x, x>k)>x^k
Polynomial Power Function
f(x)=Kx^n, K ^ n are constants, can be negative or fraction
Polynomial Functions
f(x)=anx^n+an-1x^n-1+…+a2x^2+anx+a0, a & k are constants
Sketch f(x)=x^2-x^4, f(x)=3x+x^2, e^x-x^4
Hill Function, for small & large x what does it look like
f(x)=(Ax^n)/(B+x^n)
Reason out denominator first, then simplify
For small x, f(x)=Ax^n/B
For large x, f(x)=A
Explain with words & graphs what lim(x>a^+/-) f(x)=L
Limit of f(x) as x approaches a is equal to L means f(x) is arbitrarily close to L provided x is sufficiently close to a (but not equal to a)
Limit of f(x) as x approaches a from below is equal to L is arbitrarily close to L provided x is sufficiently close to a and x<a (left hand limit)
Limit of f(x) as x approaches a from above is equal to L is arbitrarily close to L provided x is sufficiently close to a and x>a (right hand limit)
If function does not approach a single number as x approaches a, the limit does not exist (DNE)
Explain with words & graphs what lim(x>a^+/-) f(x)=+/- infinity
Vertical asymptotes
Explain with words & graphs what lim(x>+/- infinity) f(x)=L
Horizontal asymptotes.
Limit of f(x) is arbitrarily large & +/- provided x is sufficiently close to a (but not equal to a)
How to calculate limit
Continuous function’s limits computed by substituting =xaf(x)=f(a)
Not nice (0/0, infinity) function’s limits computed by simplifying into nice function, finding holes, VA & HA asymptotes & substituting
How to find features of quotient functions
(x+3)/(x+3)(x-2) Hole: x=-3, Vertical A: x=2 sub in x=1.9, 2.1 to find +/- limits, Horizontal A: n<d y=0, n=d y=coe/coe, n>d y=oblique, sub in x=+/- infinity
Identify & explain what a function with a continuous domain is
f(x) is continuous at x=a if xaf(x)=f(x), unbroken, consistent graph, no abrupt changes in value (ex. Polynomials, trig, e^x, log(x))
Discontinuous (ex. piecewise, traffic light)
Points of Discontinuity
Jump Discontinuity: piecewise
Removable Discontinuity: hole
Infinity Discontinuity: vertical asymptotes
When given a function with paraments, how to select parameter values to make a function continuous
Set equations equal at x=a, if x<k=f(x), xk=g(x), f(k)=g(k), solve for variable
Average Rate of Change
Slope of a Secant Line =[f(x1)-f(x0)]/x1-x0
Instantaneous Rate of Change
Derivative of f(x) at x0 is f’(x0)=d/dxf(x)=lim(h>0)[f(x+h)-f(x)]/h
Doesn’t exist at corners, sudden changes in slope, jumps or where the limit is infinite number
Solve for f’(1), if f(x)=x^2
Linear Approximation + Use
Linear Approximation/Tangent Line to f(x) at x=a is L(x)=f(a)+f’(a)(x-a), used to approximate hard values of functions
Linear approximation to f(x)=e^x at x=a=0, f’(0)=1
Find e^1/10
L(x)=f’(0)+f’(0)(x-0)=e0+1(x-0) = 1+x
L(1/10)=1+1/10=1.1
Euleur’s Number e
lim(h>0) eh-1/h=1
Solve for derivative of e^x
& e^f(x)
e^x
e^f(x)*f’(x)
Differential Equation + 2 Examples
Differential Equation y’(t)=y(t): a unknown function & its derivatives with a solution that satisfies it
- y=ex, y’=ex, velocity vs acceleration, compound interest, bacteria growth, population growth
- y=Cex, y’=Cex, h0Ce(x+h)-Cexh=h0Cexeh-Cexh=h0Cex(eh-1)h=Cex
Linearity of Differentiation + Proof
Proof using the limit definition of derivative & use to break down derivatives of sums + constant multiples
Power Rule + Proof
Proof using the limit definition of derivative & use for integer exponents
Derivative of a function
Find derivative of f(x)=3x+x^2
Derivative of an exponent (a^x) & tan(x)
d/dx 4^x=4^xlog(4)
d/dx tan(x)=sec2(x)
Reciprocal Rule + Proof
Product Rule + Proof
Find derivative of f(x)=x^2+1/x^2
Find derivative of f(x)=x*e^x
Quotient Rule + Proof
Find derivative of f(x)=(x^3+1)(x^2+2)
Graphically show trig f(x)s & their derivatives
Addition Formulas
sin(a+b)=sin(a)cos(b)+sin(b)cos(a)
cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
limh>0 sin(h)/h=1
limh>0 cos(h)-1/h=0
Derivative sinx & cosx proof
