Calculus 1

What are the ways limits are computed algebraically?

Plug in, Expanding/Factoring/Combining, Conjugate, Special limits, Piecewise Limits, Thinking Limits

What is the “Plug in” method for computing limits?

If one can ‘plug in’ the limiting value, then the limit is that value (since the function we give you are often continuous)

• As long as it is not piecewise

What is the Expanding/Factoring/Combining method for computing limits?

When one obtains 0/0, often expanding or factoring will cancel the terms causing the ‘problem’ [Note: When allowed, one could use l’Hopital’s Rule]

What is the Conjugate method for computing limits?

When doing limits involving roots, often multiplication by 1 using the conjugate is the trick, i.e. change the sign on the radical or constant and multiply by 1 using this form

What is the Special Limits method for computing limits?

Three most common special limits: lim (x=>0) sin x/x = 1, lim (x=>0) 1-cos x/x = 0, and lim (x=>infinity) (1 + 1/x)^x = e

What is it important to remember for thinking limits?

That if the limit doesn’t state if you are finding the LHL and the RHL, then you must find both. If the LHL and RHL do not match then its DNE

What is the Rational Limits method for computing limits?

A rational function is a ratio of polynomial, i.e. p(x)/q(x) where p,q are polynomials. Remember the degree of a polynomial is the largest power of x that appears.

• If deg q > deg p then lim = 0

• If deg p > deg q then lim = + or - infinity

• If deg p = deg q then lim = ratio leading coefficients

To verify the correct answer multiply the top and bottom by 1/x^degree denominator

What is the definition of Continuous?

A function f(x) is continuous at x=r if… f( r) = lim [x=>r] f(x)

• We say f(x) is continuous on an interval I if it is continuous for all x - values in I

• If f(x) is not continuous at x = r or on I, we say f(x) is discontinuous at x=r or on I

  • Removable Discontinuity - lim [x=>r] f(x) exists and F ( r) is not defined (there’s a hole)

  • Jump Discontinuity - F ( r) does not equal lim [x=>r] or lim [x=>r^-1] f(x) and lim[x=>r^+] f(x) exist but lim [x=>r] f(x) = DNE

  • Infinite Discontinuity - lim [x=>r^-] f(x) = + or - infinity or lim [x=>r^+] f(x) = + or - infinity

What kinds of functions are continuous?

Polynomials, Exponential functions, Logarithmic functions (everywhere they are defined), Rational functions(everywhere they are defined), Sin x and Cos x, and Power functions (whenever defined)

First 5 derivatives

• d/dx (constant) = 0

• d/dx xⁿ = n*x^n-1

• d/dx #^x = #^x ln #

• d/dx e^x = e^x

• d/dx log_b(x) = 1 / (x) ln b

Second 5 derivatives

• d/dx ln (x) = 1 / (x)

• d/dx sin (x) = cos (x)

• d/dx cos (x) = - sin (x)

• d/dx tan (x) = sec²(x)

• d/dx csc (x) = - csc(x) cot(x)

Third 5 derivatives

• d/dx sec(x) = sec(x) tan (x)

• d/dx cot (x) = -csc²(x)

• d/dx arcsin (x) = 1/ square root (1 - (x)²)

• d/dx arccos (x) = -1/ square root (1 - (x)²)

• d/dx arctan (x) = 1/ 1 + (x)²

Fourth 3 derivatives

• d/dx arccsc (x) = -1/ lxl square root ((x)²-1)

• d/dx arcsec (x) = 1/ lxl square root ((x)²-1)

• d/dx arccot (x) = -1/ 1+(x)²

f’(a) =

lim [h=>a] f(a+h) - f(a)/h

• Derivatives gives ROC

• Derivatives gives slope of tangent line

Chain Rule Derivatives

d/dx (F(g(x))) = f’(g(x)) * g’(x)

Product Rule Derivatives

d/dx (F g) = f’ g + f g’

Quotient Rule Derivatives

d/dx (F/g) = f’ g - g’ f / g²