Calc BC Studying

Question 1

Critical points

Answer 1

Where the first derivative equals 0.

Question 2

Extreme value Theorem

Answer 2

If a function f is continuous on a closed interval [a,b] , then f has both a maximum and a minimum on the interval.

Question 3

Mean Value Theorem

Answer 3

If a function f(x) is continuous on a closed interval [a,b] and differentiable on at least the open interval then f’(c) = f(b)-f(a) / b-a somewhere at x = c in the interval (a,b)

Question 4

first derivative test

Answer 4

Determine the increasing/decreasing behaviors on each side of each candidate including the endpoints using the first derivative

Question 5

How to locate inflection points

Answer 5

Where the second derivative changes signs

Question 6

Second derivative test

Answer 6

Determine the sign of the second derivative f’‘(x) at each candidate location x = c,
if f”(c) >0, the f is concave up.
Thus, f has a minimum

if f”(c) < 0, the f is concave down.
Thus, f has a maximum

if f”(c) =0 the test fails

Question 7

Uses second derivatives for estimates

Answer 7

f”(x)<0 concave down so underestimate
f”(x)>0 concave up so overestimate

Question 8

Riemann sums

Answer 8

1. Divide the base into n congruent intervals and calculate the common length.
2. Label the x-value of each right endpoint with its position. Use a general expression in terms of i, where i = 1, 2, 3,…, n
3. Substitute this general expression into f(x) to calculate the rectangle heights.
4. Multiply this new expression (for the rectangle heights) by the common base length (from step 1) to get rectangle areas.
5. Add these up. Use E notation.
6. Simplify the sum. Use the formulas on (p. 260), provided.
7. Evaluate the limits as n → infinity.

Question 9

Average velocity

Answer 9

integral of v(t) dt from a to b divided by b-a

Question 10

Final position

Answer 10

inital position, f(a), plus the total change in position, integral a to b for f’(x).

Question 11

distance

Answer 11

Absolute value of velocity

Question 12

Anti derivative of 1/x

Answer 12

ln|x|+c

Question 13

Anti derivative of 1/x

Answer 13

ln|x|+c

Question 14

Anti derivative of tan

Answer 14

-ln|cosx|+c

Question 15

Anti derivative of sec x

Answer 15

ln|secx+tanx|+c

Question 16

Anti derivative of sin x

Answer 16

-cosx +c

Question 17

Anti derivative of cos x

Answer 17

sin x + C

Question 18

Anti derivative of cot x

Answer 18

ln|sin x| +c

Question 19

Anti derivative of csc x

Answer 19

-ln|csc x + cot x|+c

Question 20

strictly monotonic

Answer 20

if it is either increasing on its entire domain or decreasing on its
entire domain.

Question 21

d/dx (ax)

Answer 21

a^x * ln a

Question 22

Log base a(x) =

Answer 22

lnx/lna

Question 23

derivative of arc sin x

Answer 23

1/√1-x^2

Question 24

derivative of arc tan

Answer 24

1/x^2 +1

Question 25

derivative of arc sec

Answer 25

1/x√x^2-1

Question 26

derivative of arc cos

Answer 26

-1/√1-x^2

Question 27

derivative of arc cot

Answer 27

-1/x^2+1

Question 27

derivative of arc csc

Answer 27

-1/x√x^2-1

Question 28

Model for exponential growth and decay

Answer 28

y=Ae^kt A is the initial amount k is the proportionality constant growth when k>0 decay when k<0

Question 29

Model for logistic growth and decay

Answer 29

y = L/1+be^-kt L is the carrying capacity (limit for all positive initial conditions) y inital = L/1+b is the initial population

Question 30

Arc length

Answer 30

integral of √ 1+f'(x)^2 dx

Question 31

l'Hopital's rule

Answer 31

the limit at a of f(x)/g(x) = 0/0 or infinity/infinity equals f'(x)/g'(x)

Question 32

nth term test

Answer 32

if the limit as n approaches infinity for a series does not equals 0, then the series a diverges

Question 33

Integral test

Answer 33

If the series a equals f(n), then the series a and the integral to infinity converge and diverge. if the integral diverges then the series diverges and if integral converges then the series converges

Question 34

p-Series

Answer 34

series of 1/n^p where p is positive and constant

Question 35

Direct comparison test

Answer 35

if 0

Question 36

Alternating series error bound

Answer 36

When approximating the sum of a convergent alternating series (so that the terms decrease in magnitude towards zero), the error is never more then the size of the first neglected term.

Question 37

conditional vs absolute converges

Answer 37

A series converges absolutely if series|a| converges. o If a series converges absolutely, then series(a) also converges. o If a series converges absolutely, then it behaves like a finite sum. * A series converges conditionally if series(a) converges but series|a| does not.

Question 38

nth Taylor polynomial for f at c

Answer 38

f^(')(c)*(x-c)^n/n!

Question 39

Alternating series error bound

Answer 39

When approximating the sum of a convergent alternating series (so that the terms decrease in magnitude towards zero), the error is never more than the size of the first neglected term. Caution! Be sure to acknowledge the conditions by writing, “Since the terms alternate while decreasing in size towards 0,…”