Math - things to remember (practice)

Question 1

Strategy to solve related rates

Answer 1

1. Draw picture of the problem
2. Write equation that relates quantities
3. Take derivative with respect to time of both sides
4. Solve for variable you need

Question 2

Linear Approximation?

Answer 2

y = f(a) + f ′(a)(x − a). first x is the actual x point, a is the a point closest to it. Afterwards, input the value of the real x for the function and see how close it was

Question 3

Differential formula?

Answer 3

dy = f ′(x)dx.

Question 4

Max & Min?

Answer 4

1. Take derivative of function and set equal to zero. Points found are critical points.
2. Input critical points into function. Larger point will be max, smaller will be minimum (if only 1 point, input 2 other x values smaller and larger than the critical point; if both y values are smaller, you have max, if they’re larger, than its a minimum)

Question 5

1st derivative test

Answer 5

i. If f ′ changes sign from positive when x < c to negative when x > c, then f(c) is a local maximum of f.
ii. If f ′ changes sign from negative when x < c to positive when x > c, then f(c) is a local minimum of f.
iii. If f ′ has the same sign for x < c and x > c, then f(c) is neither a local maximum nor a local minimum of

Question 6

Concavity?

Answer 6

If f ′ is increasing over I, we say f is concave up over I. If f ′ is decreasing over I, we say f is concave down over I.

Question 7

How do you test for concavity?

Answer 7

i. If f ″(x) > 0 for all x ∈ I, then f is concave up over I.
ii. If f ″(x) < 0 for all x ∈ I, then f is concave down over I.

Question 8

Local max/min?

Answer 8

i. If f ″(c) > 0, then f has a local minimum at c.
ii. If f ″(c) < 0, then f has a local maximum at c.
iii. If f ″(c) = 0, then the test is inconclusive.

Question 9

L’Hopital’s Rule?

Answer 9

plug in limit value of f(x)/g(x) = 0/0, then do f’(x)/g’(x) and plug in

Question 10

limit definition of derivative?

Answer 10

F(x+h) - F(x)/h

Question 11

More derivatives

Answer 11

log b X = 1/x ln b
e^x = e^x
e^2x = 2e^2x
√x = 1/2√x
ln x = 1/x
ln 2x = 2/2x
ln e = 1

Question 12

trig derivatives

Answer 12

sin x = cos x
cos x = -sin x
tan x = sec^2 x
csc x = csc cot x
sec x = sec tan x
cot x = csc^2 x

Question 13

chain, product, power, quotient rule

Answer 13

power x^n = nx^n-1

product f(x) g(x) = f’(x) g(x) + f(x) g’(x)

quotient f(x)/g(x) = f’(x) g(x) - f(x) g’(x)
g’(x)

chain rule = f(g(x)) = f’(g(x) g’(x)

Question 14

Definite integral vs indefinite

Answer 14

definite has an upper and lower limit; indefinite does not

Question 15

Integration of definite?

Answer 15

a numerical value, determined by F(b) - F(a). No constant

Question 16

Integration of Indefinite?

Answer 16

F(x) + C

Question 17

Area between (two) curves

Answer 17

Area = integral from a to b of f(x) d(x) minus g(x) d(x)

( a double definite integral)

Question 18

Disk method?

Answer 18

Volume = integration from b minus a of pi r^2 x dx

Question 19

washer method?

Answer 19

volume = pi x the square root of y^2 at d minus c