AP CALC

Question 1

Trig derivatives can be derived through

Answer 1

quotient rule, sinx and cos x

Question 2

Quotient rule derives the ___ first, then the ___.

Answer 2

numerator, denominator

Question 3

Inverse function derivatives can be derived through

Answer 3

chain rule / implicit diff on f(f^-1(x)) = x

Question 4

Derivative of f^-1 (x) is

Answer 4

1/f’(f^-1(x))

Question 5

The linearization of f(x) at point x=a is equivalent to the equation of

Answer 5

the tangent line at a

Question 6

state the conditions of the MVT, the theorem, and the application

Answer 6

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)

Question 7

state the conditions of the EVT and the theorem

Answer 7

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

Question 8

state the conditions of the IVT and the theorem

Answer 8

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

Question 9

explain the first derivative test for local extrema and conditions

Answer 9

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.

Question 10

explain the candidates test for absolute extrema and conditions

Answer 10

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.

Question 11

explain concave up and concave down with graph representations and tangent lines and second derivatives

Answer 11

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.

Question 12

explain the second derivative test for local extrema and the conditions

Answer 12

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.

Question 13

Explain left and right reimann sums from x=a to b.

Answer 13

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.

Question 14

Explain the midpoint reimann sum.

Answer 14

Still x=(b-a)/n yields n rectangles, take the midpoint of every two consecutive points and that defines the height.

Question 15

Explain the trapezoidal sum.

Answer 15

Still x=(b-a)/n yields n rectangles, take the area of the trapezoid between every two consecutive points.

Question 16

Describe how to find whether reimann sums, midpoint sums, and trapezoidal sums are over or underestimates of the actual area.

Answer 16

Reimann sums = increasing or decreasing, midpoint and trapezoidal = concave up or down. simply draw example diagrams to remember which is which.

Question 17

What are critical numbers?

Answer 17

x=c where f’(c)=0 or DNE

Question 18

What are points of inflection?

Answer 18

Where f’‘(x)=0 or DNE and f’‘(x) changes sign.

Question 19

State the logistic model general equation and how to derive the solutions

Answer 19

dy/dt = ky(a-y), solutions can be derived via separable equations

Question 20

u8

Question 21

First derivative of parametric equation is

Answer 21

(dy/dt)/(dx/dt)

Question 22

Second derivative of parametric equation can be derived by

Answer 22

inputting dy/dx=y into formula dy/dx = (dy/dt) / (dx/dt)

Question 23

Rewrite d^2y/dx^2

Answer 23

d(dy/dx)/dx

Question 24

Explain how to derive the parametric arc length formula

Answer 24

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

Question 25

Explain how to derive the polar derivatives and how to regard r and theta.

Answer 25

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.

Question 26

Explain how to derive the polar arc length.

Answer 26

Use parametric arc length and input polar derivatives.

Question 27

Will you remember to add the r when integrating in polar coordinates?

Answer 27

Yes.

Question 28

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

Answer 28

pi/6 switches

Question 29

State the bounded-monotonic theorem and the specific cases.

Answer 29

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.

Question 30

State the test for divergence/nth term test

Answer 30

If $\lim_{n \to \infty} a_n \ne 0,$ the series diverges. It follows that if the series converges, $\lim_{n \to \infty} a_n = 0.$

Question 31

State the absolute value theorem for limits.

Answer 31

If limit n->inf |a_n| = 0 then limit n->inf a_n = 0

Question 32

State the integral test and the conditions on the series

Answer 32

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 $[1, \infty).$)

Question 33

State the direct comparison test and conditions.

Answer 33

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.

Question 34

State the limit comparison test and conditions.

Answer 34

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.

Question 35

State the alternating series test and conditions.

Answer 35

An alternating series $\sum_{n=1}^\infty (-1)^{n+1} b_n$ is convergent if $b_{n+1} \le b_n$ (“decreasing”) and $\lim b_n = 0.$

Question 36

State the ratio test and root test and what they test for.

Answer 36

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).

Question 37

Explain how to find the radius and interval of convergence.

Answer 37

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 $\lvert{r}\rvert < 1).$

Question 38

State the alternating series estimation theorem and the conditions

Answer 38

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.

Question 39

Explain how to derive taylor series.

Answer 39

f(0) = a_0, repeatedly differentiate then take f^(n)(0)

Question 40

What is the derivative of sin^-1(x)

Answer 40

1/sqrt(1-x^2)

Question 41

What is the derivative of cos^-1(x)

Answer 41

-1/sqrt(1-x^2)

Question 41

What is the derivative of tan^-1(x)

Answer 41

1/(1+x^2)

Question 42

explain why a power series can only be conditionally convergent on the endpoints.

Answer 42

Elsewhere it has to be absolutely convergent, because the ratio test tests for ABSOLUTE CONVERGENCE!

Question 43

state taylor’s inequality and the definitions

Answer 43

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 $\lvert x-a \rvert \le d.$