Calc midterm flashcards

derivative of sinx

cosx

derivative of cosx

-sinx

derivative of tanx

sec²x

derivative of secx

secxtanx

derivative of cscx

-cscxcotx

derivative of cotx

-csc²x

sin2x?

2sinxcosx (the same as the derivative of sin²x)

cos2x?

1. 2cos²x-1

2. 1-2sin²x

   1. cos²x-sin²x

cos²x-sin²x?

cos2x

cos²x+sin²x?

1

sin²x-cos²x?

-cos2x

ln(1)?

0

e⁰

1

e¹

2.7

derivative of ln(x)?

1/x

derivative of sin^-1

1/sqrt(1-x²)

derivative of cos^-1

-1/sqrt(1-x²)

derivative of tan^-1

1/(x²+1)

derivative of of tan^-1(x²)?

1. Implement chain rule

1/((x²)²+1) * 2x

derivative of e^x

e^x

derivative of log_bx?

1/xlnb

^Notice the b comes out

How to rewrite log_bx in ln?

ln(x)/ln(b)

domain and range of ln

domain: [1, +inf)

range: (-inf, +inf)

Domain and range of e?

Domain: (-inf, +inf)

Range: (0, +inf)

Domain and range of log_bx?

domain: (0, +inf)

range: (-inf, + inf)

What is the exception of log_bX?

b > 0, and b ≠ 1

cos(0)?

1

cos(pi)?

-1

cos(2pi)?

1

cos(-pi)?

-1

cos(3pi/2)?

0

3pi/2 in degrees?

270

sin(0)?

0

sin(pi)?

0

sin(3pi/2)?

-1

sin(-pi)?

0

what is the formula for reference angle in quadrant IV?

2pi - reference anglewhat is the formula for reference angle in quadrant

what is the formula for reference angle in quadrant I?

reference angle

what is the formula for reference angle in quadrant II?

pi - reference angle

what is the formula for reference angle in quadrant III?

pi + reference angle?

What should you do if you have a negative angle?

Add 2pi, until posiive

sinx = 1/2

x = pi/6

cos x = √(2)/2

x = pi/4

cos x = 1/2

x = pi/3

cos x = √(3)/2?

x = pi/6

sin x = √(2)/2

x = pi/4

sin x = √(3)/2?

x = pi/3

180/3?

60

180/6?

30 degrees

√a*√a = ?; what about √a/a

a,

becase a^1/2a^1/2= a^{1/2 + 1/2} =a¹

^when √a/a, the answer is 1/√a

If I have x² ≥ 2, what are my two answers from that (remember inequalities are involved)

x < -√2 AND x > √2

WHENEVER YOU HAVE x² > a (a > 0) → x < -√a AND x > √a

If I have x² ≤ 2, what are my two answers from that (remember inequalities are involved)

-√2 < x < √2

tan2x?

2tanx/1-tan²x

sec²x?

tan²x+1

csc²x?

1+cot²x

tan²+1?

sec²x

1+cot²x?

csc²x

What is the reference angle for -pi/3?

5pi/3

-pi/3 + 2pi (6pi/3) = 5pi/3

What do I put in an answer if I’m meant to find ALL the solutions?

You add “+ 2kpi”

If I have cos (x + 3pi/3) = √2/2 and I want to find ALL the solutions, what is my FIRST step?

Make u = x + 3pi/3 AND u = x - 3pi/3

If i have a*b ≤ 0, what does that mean for a and b?

either 1. a = (-) and b = (+). or 2. a = (+) and b = (-)

If I have cos ≤ 1/2, what is my strategy?

Try to draw out when cos 1/2?

What is cos graph?

sin graph

Restrictions for cos^-1?

domain: [-1,1]

range: (0, pi)

restrictions for sin^-1?

domain: [-1,1]

range: -pi/2, pi/2

tan-1 restrictions?

domain: (-inf, +inf)

range: [-pi/2, pi/2]

tanx restrictions?

• In plain words: All real numbers except odd multiples of pi/2 (such as pi/2, 3pi/2, -pi/2

• Why: The tangent function is sinx/cosx It is undefined wherever cos(x) = 0

tan x = 0

x = 0 → because sinx/cosx = 0/1 = 0

tan x = 1/2

around 26 degrees, what matters is tan x = ½ is NOT a special valueSp

Special values of tan?

1. √3

2. 1

3. √3/3

tan x = √3/3

pi/6

tan x = 1

pi/4

sin(pi/4)/cos(pi/4)?

1 → because at pi/4, sin and cos are the same value

tan x = √3

x = pi/3

b⁰?

1

b^x, domain and range?

domain: (-inf, +inf)

range: [0, +inf)

how to do change base to e trick?

b^x= e^xln(b)

domain and range of √x

domain: [0, +inf)

range: [0, inf)

what is special about x^x

Has a variable base and exponent, neither are a constant

how to make 5^cos(x) into change base to e trick?

e^cos(x)ln(5)

how to derive x^x

x^x(1+ln(x))

lim of 0⁰

This limit DNE

range and domain of ln(x+1)

domain: (-1, + inf)

range: (-inf, + inf)

Range of b^x

range: [0, +inf)

ln(e)?

1

what is a natural domain

When it’s only + numbers

When is something a function?

If it passes the vertical line test (only 1 y value per x); one to one

When is a function invertible?

If it passes the horizontal line test (only 1 x value per y)

What is a common function that is NOT invertible

X²

if I have function f(x) = √x+2) + 5, is there an x-intercept (when y = 0)?

NO, because the range of y is y >= 5 (look at domain and range rules if confused)

When are functions parallel

If they share the same slope

Definition of a power function

xⁿ, where n is a CONSTANT

When is cotangent undefined?

at 0, and pi. Whenever sin = 0. Because cotangent = cosx/sinx

what is the natural log?

ln(x) = log_ex

The inverse of e?

ln

how to find the domain of an inverse function?

The domain of an inverse, will be the range of the original and vice versa!

Why is the definition of a domain important in inverses?

because in inverse function you need to be able to reverse, every single INPUT of the function as an inverse. REMEMBER the range of the og needs to match the domain of the new, vice versa. So the domain is important, it will tell you if a function is actually the inverse

If lim x→a f(x) exists, then is f(x) defined at a?

No it doesn’t have to be. Example is removeable discontinuity

If lim x→ a f(x) = +inf, then does that mean the limit exists and it is an infinity?

No because this is short hand for the limit DNE, because in the definition of a limit, for a limit to exist it MUST be approaching a finite number