AP CALC Flashcards
Trig derivatives can be derived through
quotient rule, sinx and cos x
Quotient rule derives the ___ first, then the ___.
numerator, denominator
Inverse function derivatives can be derived through
chain rule / implicit diff on f(f^-1(x)) = x
Derivative of f^-1 (x) is
1/f’(f^-1(x))
The linearization of f(x) at point x=a is equivalent to the equation of
the tangent line at a
state the conditions of the MVT, the theorem, and the application
conditions are f is continuous on [a,b] and differentiable on (a,b), then there exists some c in (a,b) such that f’(c) = (f(b)-f(a)) / (b-a).
can be applied to maximize a function given a derivative, and the obvious finding of existence of f’(c)
state the conditions of the EVT and the theorem
if f is continuous on [a,b] then there is a max and min value (i.e. there exists c such that f(c) ≥ f(x) for all x, vice versa) intiuitively true because if not continous could have vertical asymptote or smth
state the conditions of the IVT and the theorem
if f is continuous over [a,b] and L lies between f(a) and f(b) then there exists c in (a,b) such that f(c)=L
explain the first derivative test for local extrema and conditions
if f is continuous and differentiable, and c is a critical point (f’(c) = 0), we look at whether the derivative changes from positive to negative or vice versa to determine if it is a local max, min, or neither.
explain the candidates test for absolute extrema and conditions
f defined on a closed interval (may not be diff. bc critical point could be dne); only critical points and endpoints can be absolute extrema. compare the values of f across all plus endpoints. the highest is local max lowest is local min.
explain concave up and concave down with graph representations and tangent lines and second derivatives
concave up: U, above all tangent lines, slope increasing, f’‘(x)>0
concave down: n, below all tangent lines, slope decreasing, f’‘(x)<0. if f’‘(x) = 0 then inflection point.
explain the second derivative test for local extrema and the conditions
if f’(c) = 0 and f’‘(c) exists: if f’‘(c)>0 then a minimum at c, if f’‘(c)<0 then a maximum at c. if =0 then no conclusion.
Explain left and right reimann sums from x=a to b.
The left sum starts at x=a, ends at one point from x=b, whereas the right sum starts at the next point over, ends at x=b. in other words, rectangles’ height is determined by their left vs right point.
Explain the midpoint reimann sum.
Still x=(b-a)/n yields n rectangles, take the midpoint of every two consecutive points and that defines the height.
Explain the trapezoidal sum.
Still x=(b-a)/n yields n rectangles, take the area of the trapezoid between every two consecutive points.
Describe how to find whether reimann sums, midpoint sums, and trapezoidal sums are over or underestimates of the actual area.
Reimann sums = increasing or decreasing, midpoint and trapezoidal = concave up or down. simply draw example diagrams to remember which is which.
What are critical numbers?
x=c where f’(c)=0 or DNE
What are points of inflection?
Where f’‘(x)=0 or DNE and f’‘(x) changes sign.
State the logistic model general equation and how to derive the solutions
dy/dt = ky(a-y), solutions can be derived via separable equations
u8
First derivative of parametric equation is
(dy/dt)/(dx/dt)
Second derivative of parametric equation can be derived by
inputting dy/dx=y into formula dy/dx = (dy/dt) / (dx/dt)
Rewrite d^2y/dx^2
d(dy/dx)/dx
Explain how to derive the parametric arc length formula
Use the substitution dy/dx into general arc length= (dy/dt)/(dx/dt) then multiply by dx/dt -> bring it out changes dx to dt
Explain how to derive the polar derivatives and how to regard r and theta.
Regard r as f(theta) and theta as the parameter. Simply use parametric derivative dy/dx = (dy/dtheta)/(dx/dtheta) and y=rsin theta, r cos theta + product rule.
Explain how to derive the polar arc length.
Use parametric arc length and input polar derivatives.
Will you remember to add the r when integrating in polar coordinates?
Yes.
How do you integrate the rose with 3 loops r=cos(3theta)? (specifically, bounds on theta?) explain how the point travels along the curve as theta =0->2pi
pi/6 switches
State the bounded-monotonic theorem and the specific cases.
If a sequence is bounded both ways and monotonic, it is convergent. More specifically, a sequence that is increasing and bounded above is convergent. Likewise, a decreasing sequence that is bounded below is convergent.
State the test for divergence/nth term test
If $\lim_{n \to \infty} a_n \ne 0,$ the series diverges. It follows that if the series converges, $\lim_{n \to \infty} a_n = 0.$
State the absolute value theorem for limits.
If limit n->inf |a_n| = 0 then limit n->inf a_n = 0
State the integral test and the conditions on the series
A series s is convergent if and only if $\int_1^\infty s_n dn$ converges. ($s_n$ must be a continuous, positive, ultimately decreasing (consider derivative) function on $[1, \infty).$)
State the direct comparison test and conditions.
Both series must be NONNEGATIVE.
If a series is always less than a convergent series, it converges. If a series is always greater than a divergent series, it diverges.
State the limit comparison test and conditions.
For two series $\sum a_n$ and $\sum b_n,$ if they are both NONNEGATIVE and lim_{n\to \infty} \frac{a_n}{b_n}$ is FINITE and POSITIVE, either both series converge or diverge.
State the alternating series test and conditions.
An alternating series $\sum_{n=1}^\infty (-1)^{n+1} b_n$ is convergent if $b_{n+1} \le b_n$ (“decreasing”) and $\lim b_n = 0.$
State the ratio test and root test and what they test for.
ratio test: If $\lim{n \to \infty} \lvert \frac{a_{n+1}}{a_n} \rvert = L < 1,$ then $\sum a_n$ is ABSOLUTELY convergent thus convergent. If $L > 1,$ it diverges. $L=1$ is inconclusive (notably for p-series/anything that fails the ROT).
$\mathbf{ROT:}$ If $\lim{n \to \infty} \sqrt[n]{\lvert a_n \rvert} = L < 1,$ then $\sum a_n$ is ABSOLUTELY convergent thus convergent. If $L > 1,$ it diverges. $L=1$ is inconclusive (notably for anything that fails the RAT).
Explain how to find the radius and interval of convergence.
RADIUS OF CONVERGENCE: For some power series, there exists a positive $R$ such that the series converges if $\lvert x-a \rvert < R$ and diverges if $\lvert x-a \rvert > R.$ To find this, use the ratio test (or root test).
INTERVAL OF CONVERGENCE: Find the radius first, then input $x=a-R$ and x = $a+R$ to evaluate convergence at endpoints of $(a-R, a+R).$ Alternatively, the function can be manipulated into a sum of geometric series formula (with $\lvert{r}\rvert < 1).$
State the alternating series estimation theorem and the conditions
For a convergent alternating series s=\sum (-1)^{n+1}b_n, |R_n| = |s-s_n| ≤ b_{n+1} where R_n is the remainder.
Explain how to derive taylor series.
f(0) = a_0, repeatedly differentiate then take f^(n)(0)
What is the derivative of sin^-1(x)
1/sqrt(1-x^2)
What is the derivative of cos^-1(x)
-1/sqrt(1-x^2)
What is the derivative of tan^-1(x)
1/(1+x^2)
explain why a power series can only be conditionally convergent on the endpoints.
Elsewhere it has to be absolutely convergent, because the ratio test tests for ABSOLUTE CONVERGENCE!
state taylor’s inequality and the definitions
TAYLOR’S INEQUALITY: Let $R_n$ be the remainder of $f(x) - T_n(x).$ If $\lvert f^{(n+1)}(x) \rvert \le M$ for $\lvert x-a \rvert \le d,$ we have $\lvert R_n(x) \rvert \le \frac{M}{(n+1)!} \lvert x-a \rvert^{n+1}$ for $\lvert x-a \rvert \le d.$
