APMA 1090 Final 2025/2026 Exam
Comprehensive Questions and Verified
Answers | Accurate Solutions | Get it
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indeterminate forms - 🧠 ANSWER ✔✔when f(x) isn't continuous at c, may
result in 0/0...etc, need to rewrite to solve
difference quotient - 🧠 ANSWER ✔✔f(x+h)-f(x)/h, used to find the slope of
the tangent line
heaviside function - 🧠 ANSWER ✔✔H(x) = {0 x < 0, 1 x >= 0}
squeeze theorem - 🧠 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 - 🧠 ANSWER ✔✔f(x) is continuous at x = a if limx->a
f(x) = f(a)
types of discontinuities - 🧠 ANSWER ✔✔removeable (hole in graph), jump
(hole but a point somewhere else), infinite (graph approaches infinity)
1
COPYRIGHT@SARAHROSAPERAL 2025/2026. YEAR PUBLISHED 2025. COMPANY REGISTRATION NUMBER: 619652435. TERMS OF USE.
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, intermediate value theorem - 🧠 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 - 🧠 ANSWER ✔✔(e^x + e^-x)/2
sinh - 🧠 ANSWER ✔✔(e^x - e^-x)/2
derivative of tanx/cotx - 🧠 ANSWER ✔✔sec^2x/-csc^2x
derivative of secx/cscx - 🧠 ANSWER ✔✔secxtanx/-cscxcotx
implicit derivatives - 🧠 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 - 🧠 ANSWER ✔✔1/(1-x^2)^(1/2)
derivative of arccosx/cos^-1x - 🧠 ANSWER ✔✔-1/(1-x^2)^(1/2)
derivative of arctanx/tan^-1x - 🧠 ANSWER ✔✔1/(1+x^2)
derivative of arcsecx/sec^-1x - 🧠 ANSWER ✔✔1/(|x|*(x^2-1)^(1/2))
derivative of logb(x) - 🧠 ANSWER ✔✔1/xln(b)
2
COPYRIGHT@SARAHROSAPERAL 2025/2026. YEAR PUBLISHED 2025. COMPANY REGISTRATION NUMBER: 619652435. TERMS OF USE.
PRIVACY STATEMENT. ALL RIGHTS RESERVED
Comprehensive Questions and Verified
Answers | Accurate Solutions | Get it
100% Correct!! | Already Graded A+
indeterminate forms - 🧠 ANSWER ✔✔when f(x) isn't continuous at c, may
result in 0/0...etc, need to rewrite to solve
difference quotient - 🧠 ANSWER ✔✔f(x+h)-f(x)/h, used to find the slope of
the tangent line
heaviside function - 🧠 ANSWER ✔✔H(x) = {0 x < 0, 1 x >= 0}
squeeze theorem - 🧠 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 - 🧠 ANSWER ✔✔f(x) is continuous at x = a if limx->a
f(x) = f(a)
types of discontinuities - 🧠 ANSWER ✔✔removeable (hole in graph), jump
(hole but a point somewhere else), infinite (graph approaches infinity)
1
COPYRIGHT@SARAHROSAPERAL 2025/2026. YEAR PUBLISHED 2025. COMPANY REGISTRATION NUMBER: 619652435. TERMS OF USE.
PRIVACY STATEMENT. ALL RIGHTS RESERVED
, intermediate value theorem - 🧠 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 - 🧠 ANSWER ✔✔(e^x + e^-x)/2
sinh - 🧠 ANSWER ✔✔(e^x - e^-x)/2
derivative of tanx/cotx - 🧠 ANSWER ✔✔sec^2x/-csc^2x
derivative of secx/cscx - 🧠 ANSWER ✔✔secxtanx/-cscxcotx
implicit derivatives - 🧠 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 - 🧠 ANSWER ✔✔1/(1-x^2)^(1/2)
derivative of arccosx/cos^-1x - 🧠 ANSWER ✔✔-1/(1-x^2)^(1/2)
derivative of arctanx/tan^-1x - 🧠 ANSWER ✔✔1/(1+x^2)
derivative of arcsecx/sec^-1x - 🧠 ANSWER ✔✔1/(|x|*(x^2-1)^(1/2))
derivative of logb(x) - 🧠 ANSWER ✔✔1/xln(b)
2
COPYRIGHT@SARAHROSAPERAL 2025/2026. YEAR PUBLISHED 2025. COMPANY REGISTRATION NUMBER: 619652435. TERMS OF USE.
PRIVACY STATEMENT. ALL RIGHTS RESERVED
