CALCULUS EXAM COMPLETE QUESTIONS
AND VERIFIED ANSWERS GRADED A+ 2026
◉Average Rate of Change . Answer: Slope of secant line between two
points, use to estimate instantanous rate of change at a point.
◉Instantenous Rate of Change . Answer: Slope of tangent line at a point,
value of derivative at a point
◉Formal definition of derivative . Answer:
◉Alternate definition of derivative . Answer: limit as x approaches a of
[f(x)-f(a)]/(x-a)
◉When f '(x) is positive, f(x) is . Answer: increasing
◉When f '(x) is negative, f(x) is . Answer: decreasing
◉When f '(x) changes from negative to positive, f(x) has a . Answer:
relative minimum
,◉When f '(x) changes from positive to negative, f(x) has a . Answer:
relative maximum
◉When f '(x) is increasing, f(x) is . Answer: concave up
◉When f '(x) is decreasing, f(x) is . Answer: concave down
◉When f '(x) changes from increasing to decreasing or decreasing to
increasing, f(x) has a . Answer: point of inflection
◉When is a function not differentiable . Answer: corner, cusp, vertical
tangent, discontinuity
◉Product Rule . Answer: uv' + vu'
◉Quotient Rule . Answer: (uv'-vu')/v²
◉Chain Rule . Answer: f '(g(x)) g'(x)
◉y = x cos(x), state rule used to find derivative . Answer: product rule
◉y = ln(x)/x², state rule used to find derivative . Answer: quotient rule
, ◉y = cos²(3x) . Answer: chain rule
◉Particle is moving to the right/up . Answer: velocity is positive
◉Particle is moving to the left/down . Answer: velocity is negative
◉absolute value of velocity . Answer: speed
◉y = sin(x), y' = . Answer: y' = cos(x)
◉y = cos(x), y' = . Answer: y' = -sin(x)
◉y = tan(x), y' = . Answer: y' = sec²(x)
◉y = csc(x), y' = . Answer: y' = -csc(x)cot(x)
◉y = sec(x), y' = . Answer: y' = sec(x)tan(x)
◉y = cot(x), y' = . Answer: y' = -csc²(x)
◉y = sin⁻¹(x), y' = . Answer: y' = 1/√(1 - x²)
AND VERIFIED ANSWERS GRADED A+ 2026
◉Average Rate of Change . Answer: Slope of secant line between two
points, use to estimate instantanous rate of change at a point.
◉Instantenous Rate of Change . Answer: Slope of tangent line at a point,
value of derivative at a point
◉Formal definition of derivative . Answer:
◉Alternate definition of derivative . Answer: limit as x approaches a of
[f(x)-f(a)]/(x-a)
◉When f '(x) is positive, f(x) is . Answer: increasing
◉When f '(x) is negative, f(x) is . Answer: decreasing
◉When f '(x) changes from negative to positive, f(x) has a . Answer:
relative minimum
,◉When f '(x) changes from positive to negative, f(x) has a . Answer:
relative maximum
◉When f '(x) is increasing, f(x) is . Answer: concave up
◉When f '(x) is decreasing, f(x) is . Answer: concave down
◉When f '(x) changes from increasing to decreasing or decreasing to
increasing, f(x) has a . Answer: point of inflection
◉When is a function not differentiable . Answer: corner, cusp, vertical
tangent, discontinuity
◉Product Rule . Answer: uv' + vu'
◉Quotient Rule . Answer: (uv'-vu')/v²
◉Chain Rule . Answer: f '(g(x)) g'(x)
◉y = x cos(x), state rule used to find derivative . Answer: product rule
◉y = ln(x)/x², state rule used to find derivative . Answer: quotient rule
, ◉y = cos²(3x) . Answer: chain rule
◉Particle is moving to the right/up . Answer: velocity is positive
◉Particle is moving to the left/down . Answer: velocity is negative
◉absolute value of velocity . Answer: speed
◉y = sin(x), y' = . Answer: y' = cos(x)
◉y = cos(x), y' = . Answer: y' = -sin(x)
◉y = tan(x), y' = . Answer: y' = sec²(x)
◉y = csc(x), y' = . Answer: y' = -csc(x)cot(x)
◉y = sec(x), y' = . Answer: y' = sec(x)tan(x)
◉y = cot(x), y' = . Answer: y' = -csc²(x)
◉y = sin⁻¹(x), y' = . Answer: y' = 1/√(1 - x²)
