Applying
Limits - The value of the limit?
lim (x → 0) (1 − cos(x)) / x²
what is
lim (x → 0) (1 − cos(x)) / x² = 1/2
avg and instantaneous rates of change - This tells you the slope of the tangent to a function at a point.
What is the derivative
derivative of primary trig functions - The derivative of sin(x)
what is cos(x)
first principle derivatives - The derivative of f(x)=2x3−5x+1
what is 6x - 5
first principle derivatives - The formula for finding the derivative of a function
What is f'(x) = lim h→0(x+h)−f(x) / h
kinematics - The velocity function from s(t)=3t2+2t
What is v(t)=6t+2
local and absolute max and mins - This explains the conditions needed for a point of inflection to exist.
What is the second derivative must be zero or undefined and must change sign at that point?
Derivatives of e and exponential functions - The derivative of ex
what is ex
Kinematics - A toy missile is shot into the air. Its height, in
metres, after t seconds is given by the function
h(t) = -4.9t2 + 15t + 0.4 , t>0
a) Determine the height of the missile after 2 s.
b) Determine the rate of change of the height
of the missile at 1 s and at 4 s.
What is a) 10.8m and b) 5.2m/s and -24.2m/s
Derivative rules - The rule used for differentiating a division of two functions.
What is the quotient rule
Determine the intervals where the function
f(x)=x3− 3x2 − 9x + 5
is increasing and decreasing.
Show all critical points and justify your answer using the first derivative.
What is
Increasing on (−∞,−1) and (3,∞)
Decreasing on (−1,3)
limits - What is a conjugate and how/when is it used
In algebra, particularly in rationalizing expressions, a conjugate refers to a binomial expression with the same terms but the opposite sign between them.
When there's a square root (or any irrational number) in the denominator, you multiply the numerator and denominator by the conjugate of the denominator to eliminate the radical.
Local and Absolute Max’s and Min’s - What is the difference between a local and an absolute maximum or minimum value of a function?
What is:
A local maximum/minimum is the highest or lowest point within a small interval of the graph.
An absolute maximum/minimum is the highest or lowest point on the entire graph over a given domain.
First principle derivatives- Derivative of h(x) = (2x - 5)4/ (x + 1)3
what is 8(2x - 5)3 (x + 1)- 3(2x-5)4/ (x + 1)4
first principle derivatives - The quotient rule formula.
what is f'(x) g(x) - f(x) g'(x) / g(x)2
Optimization - The dimensions of a rectangle with perimeter 20 that has max area.
What is a square: side length 5, by solving with A=x(10−x)
Curve Analysis of trig functions -
What is curve analysis in calculus, and what features are typically examined when analyzing the graph of a trigonometric function?
What is the process of examining the behavior and shape of a function's graph using derivatives and key features?
For trigonometric functions, curve analysis includes:
Amplitude
Period and frequency
Phase shift
Vertical shift
Critical points (where f′(x)=0)
Maxima and minima
Intervals of increase and decrease
Concavity and points of inflection
Uoptimization - A farmer has 1200 meters of fencing and wants to build a rectangular enclosure divided into four equal sections using three fences parallel to one of the sides (i.e., three internal dividers).
What are the dimensions of the pen that will give the maximum possible area, and what is that maximum area?
What is L = 120m and W = 300m
First principle derivatives - What does the second derivative test tell us about a critical point?
What is:
Local and absolute maxs and mins - For the quartic function defined by
f (x) = ax4 + bx2 + cx + d, find the values of
a, b, c, and d such that there is a local
maximum at (0, 6) and a local minimum
at (1, 8).
What is a = 2 b = -4 c = 0 d = -6
Derivatives of e and exponential functions -
Differentiate
f (x) = x e2x + 2e-3x
what is f′(x) = f′(x)=e2x+2xe2x − 6e-3x
Limits - Name and describe all the types of discontinuities a function can have.
What are:
Removable Discontinuity — The limit exists but the function is either not defined at that point or the function’s value is different from the limit (a “hole” in the graph).
Jump Discontinuity — The left-hand and right-hand limits exist but are not equal, causing a “jump” in the graph.
Infinite (or Essential) Discontinuity — The function approaches infinity or negative infinity near the point (vertical asymptote).
Oscillating Discontinuity — The function oscillates infinitely near the point, so the limit does not exist (e.g., sin(1/x)\sin(1/x)sin(1/x) at 0).
A cone fills with water. Given the volume function V=1/3πr2h and a relationship between r and h, find the rate of height change when volume is increasing at 2 cm³/s.
what is 8/πh2
Intervals of increase and decrease - What are the conditions that must be satisfied for a point of inflection to occur?
What is
Note: f’’(x) = 0 or DNE at the inflection point, but the sign change is what confirms it.