APMA 1090 Final
indeterminate forms - correct answer ✔✔when f(x) isn't continuous at c, may result in 0/0...etc, need to
rewrite to solve
difference quotient - correct answer ✔✔f(x+h)-f(x)/h, used to find the slope of the tangent line
heaviside function - correct answer ✔✔H(x) = {0 x < 0, 1 x >= 0}
squeeze theorem - correct answer ✔✔If f(x) <= g(x) <= h(x) for x near a, and limx->a f(x) = limx->a h(x) =
L, then limx->a g(x) = L
definition of continuity - correct answer ✔✔f(x) is continuous at x = a if limx->a f(x) = f(a)
types of discontinuities - correct answer ✔✔removeable (hole in graph), jump (hole but a point
somewhere else), infinite (graph approaches infinity)
intermediate value theorem - correct answer ✔✔if f(x) is continuous on [a, b] and N is a number in
between f(a) and f(b) then somewhere there is a point c in between a and b such that f(c) = N
cosh - correct answer ✔✔(e^x + e^-x)/2
sinh - correct answer ✔✔(e^x - e^-x)/2
derivative of tanx/cotx - correct answer ✔✔sec^2x/-csc^2x
derivative of secx/cscx - correct answer ✔✔secxtanx/-cscxcotx
, implicit derivatives - correct answer ✔✔1) take the derivative of both sides, treating y as a function of x,
2) solve for dx/dy, result will involve both x and y
derivative of arcsinx/sin^-1x - correct answer ✔✔1/(1-x^2)^(1/2)
derivative of arccosx/cos^-1x - correct answer ✔✔-1/(1-x^2)^(1/2)
derivative of arctanx/tan^-1x - correct answer ✔✔1/(1+x^2)
derivative of arcsecx/sec^-1x - correct answer ✔✔1/(|x|*(x^2-1)^(1/2))
derivative of logb(x) - correct answer ✔✔1/xln(b)
derivative of b^x - correct answer ✔✔b^x ln(b)
linearization - correct answer ✔✔L(x) = f(a) + f'(a)(x - a)
hyperbolic function derivatives - correct answer ✔✔know em
Fermat's Theorem - correct answer ✔✔if there is a local min or max at x = c, and f'(c) exists then f'(c) = 0,
this does not work the other way around
absolute mins and maxes - correct answer ✔✔occur at end points and critical numbers
critical numbers - correct answer ✔✔where f'(c) = 0 or does not exist
extreme value theorem - correct answer ✔✔if f(x) is continuous on closed interval [a, b], then f(x) has an
absolute max at f(c) and absolute min at f(d) for numbers c and d in [a, b]
indeterminate forms - correct answer ✔✔when f(x) isn't continuous at c, may result in 0/0...etc, need to
rewrite to solve
difference quotient - correct answer ✔✔f(x+h)-f(x)/h, used to find the slope of the tangent line
heaviside function - correct answer ✔✔H(x) = {0 x < 0, 1 x >= 0}
squeeze theorem - correct answer ✔✔If f(x) <= g(x) <= h(x) for x near a, and limx->a f(x) = limx->a h(x) =
L, then limx->a g(x) = L
definition of continuity - correct answer ✔✔f(x) is continuous at x = a if limx->a f(x) = f(a)
types of discontinuities - correct answer ✔✔removeable (hole in graph), jump (hole but a point
somewhere else), infinite (graph approaches infinity)
intermediate value theorem - correct answer ✔✔if f(x) is continuous on [a, b] and N is a number in
between f(a) and f(b) then somewhere there is a point c in between a and b such that f(c) = N
cosh - correct answer ✔✔(e^x + e^-x)/2
sinh - correct answer ✔✔(e^x - e^-x)/2
derivative of tanx/cotx - correct answer ✔✔sec^2x/-csc^2x
derivative of secx/cscx - correct answer ✔✔secxtanx/-cscxcotx
, implicit derivatives - correct answer ✔✔1) take the derivative of both sides, treating y as a function of x,
2) solve for dx/dy, result will involve both x and y
derivative of arcsinx/sin^-1x - correct answer ✔✔1/(1-x^2)^(1/2)
derivative of arccosx/cos^-1x - correct answer ✔✔-1/(1-x^2)^(1/2)
derivative of arctanx/tan^-1x - correct answer ✔✔1/(1+x^2)
derivative of arcsecx/sec^-1x - correct answer ✔✔1/(|x|*(x^2-1)^(1/2))
derivative of logb(x) - correct answer ✔✔1/xln(b)
derivative of b^x - correct answer ✔✔b^x ln(b)
linearization - correct answer ✔✔L(x) = f(a) + f'(a)(x - a)
hyperbolic function derivatives - correct answer ✔✔know em
Fermat's Theorem - correct answer ✔✔if there is a local min or max at x = c, and f'(c) exists then f'(c) = 0,
this does not work the other way around
absolute mins and maxes - correct answer ✔✔occur at end points and critical numbers
critical numbers - correct answer ✔✔where f'(c) = 0 or does not exist
extreme value theorem - correct answer ✔✔if f(x) is continuous on closed interval [a, b], then f(x) has an
absolute max at f(c) and absolute min at f(d) for numbers c and d in [a, b]
