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APMA 1090 Final Questions and Correct
Answers
indeterminate forms Ans: when f(x) isn't continuous at c, may
result in 0/0...etc, need to rewrite to solve
difference quotient Ans: f(x+h)-f(x)/h, used to find the slope of the
tangent line
heaviside function Ans: H(x) = {0 x < 0, 1 x >= 0}
squeeze theorem Ans: 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 Ans: f(x) is continuous at x = a if limx->a
f(x) = f(a)
types of discontinuities Ans: removeable (hole in graph), jump
(hole but a point somewhere else), infinite (graph approaches
infinity)
intermediate value theorem Ans: 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 Ans: (e^x + e^-x)/2
sinh Ans: (e^x - e^-x)/2
derivative of tanx/cotx Ans: sec^2x/-csc^2x
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derivative of secx/cscx Ans: secxtanx/-cscxcotx
implicit derivatives Ans: 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 Ans: 1/(1-x^2)^(1/2)
derivative of arccosx/cos^-1x Ans: -1/(1-x^2)^(1/2)
derivative of arctanx/tan^-1x Ans: 1/(1+x^2)
derivative of arcsecx/sec^-1x Ans: 1/(|x|*(x^2-1)^(1/2))
derivative of logb(x) Ans: 1/xln(b)
derivative of b^x Ans: b^x ln(b)
linearization Ans: L(x) = f(a) + f'(a)(x - a)
hyperbolic function derivatives Ans: know em
Fermat's Theorem Ans: 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 Ans: occur at end points and critical
numbers
critical numbers Ans: where f'(c) = 0 or does not exist
extreme value theorem Ans: 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]
© 2025 All rights reserved
APMA 1090 Final Questions and Correct
Answers
indeterminate forms Ans: when f(x) isn't continuous at c, may
result in 0/0...etc, need to rewrite to solve
difference quotient Ans: f(x+h)-f(x)/h, used to find the slope of the
tangent line
heaviside function Ans: H(x) = {0 x < 0, 1 x >= 0}
squeeze theorem Ans: 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 Ans: f(x) is continuous at x = a if limx->a
f(x) = f(a)
types of discontinuities Ans: removeable (hole in graph), jump
(hole but a point somewhere else), infinite (graph approaches
infinity)
intermediate value theorem Ans: 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 Ans: (e^x + e^-x)/2
sinh Ans: (e^x - e^-x)/2
derivative of tanx/cotx Ans: sec^2x/-csc^2x
© 2025 All rights reserved
, 2 | Page
derivative of secx/cscx Ans: secxtanx/-cscxcotx
implicit derivatives Ans: 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 Ans: 1/(1-x^2)^(1/2)
derivative of arccosx/cos^-1x Ans: -1/(1-x^2)^(1/2)
derivative of arctanx/tan^-1x Ans: 1/(1+x^2)
derivative of arcsecx/sec^-1x Ans: 1/(|x|*(x^2-1)^(1/2))
derivative of logb(x) Ans: 1/xln(b)
derivative of b^x Ans: b^x ln(b)
linearization Ans: L(x) = f(a) + f'(a)(x - a)
hyperbolic function derivatives Ans: know em
Fermat's Theorem Ans: 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 Ans: occur at end points and critical
numbers
critical numbers Ans: where f'(c) = 0 or does not exist
extreme value theorem Ans: 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]
© 2025 All rights reserved
