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Calc 140 > Chapter 3 > Flashcards

Chapter 3 Flashcards

1
Q

What is the constant rule? (DxC=?)

A

The derivative of a constant is always 0. (DxC=0)

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2
Q

What is the derivative of x? (DxX=?)

A

The derivative of x is 1. (DxX=1) (y=x)

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3
Q

What is the power rule for derivatives? (DxX^n=?)

A

The derivative of x^n is nx^(n-1). (DxX^n=nX^(n-1))

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4
Q

What is the sum/difference rule? ((f+g)prime=?)

A

The sum/difference of (f +/- g)prime is fprime +/-g prime.

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5
Q

What is the derivative of a constant times a function of x? (DxCf(x)=?)

A

The constant times the derivative of the function of x. (DxCf(x)=C*fprime(x))

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6
Q

What is the definition of the number e?

A

The number e is such that the lim h->0 (e^h-1)/h=1. (That the tangent slope of e^x at x=0 is 1)

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7
Q

What is the product rule? ((fgl*m)prime)=?

A

((fglm)prime= fprimeglm+fgprimelm+fglprimem+fgl*mprime

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8
Q

What is the quotient rule? (f/g)prime=?

A

(f/g)prime=(fprimeg-fgprime)/g^2

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9
Q

DxSinx=?

A

Cosx

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10
Q

DxCosx=?

A

-Sinx

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11
Q

DxTanx=?

A

Sec^2x

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12
Q

DxCscx=?

A

-CscxCotx

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13
Q

DxSecx=?

A

SecxTanx

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14
Q

DxCotx=?

A

-Csc^2x

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15
Q

What is lim (theta)->0 Sin(theta)/(theta)?

A

lim (theta)->0 Sin(theta)/(theta)=1

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16
Q

What is lim (theta)->0 (Cos(theta)-1)/(theta)?

A

lim (theta)->0 (Cos(theta)-1)/(theta)=0

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17
Q

What is the Chain Rule? Dx(f o g)(x)=?

A

The Chain Rule states that the derivative of (f o g)(x) is f(prime)(g(x))(gprime(x) (the derivative of the outside function with the inside function intact and still inside it multiplied by the derivative of the inside)

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18
Q

What is the derivative of e^x?

A

Dx e^x=e^x

19
Q

What is the derivative of e^f(x)?

A

Dx e^f(x)=e^f(x)*fprime(x) (The derivative of e to the function f(x) is e to the function f(x) multiplied by the derivative of the function f(x)) (This comes from the chain rule)

20
Q

What is the method for implicit differentiation? What do you use this for

A

Given an equation, find the prime (thinking of y(dx/dy) as f(x))(use the chain rule for substituting), and then solve for dx/dy. With the found equation, you can find the slope of the tangent line at a given point by plugging in values of the given point.

21
Q

DxSin^-1(x)=?

A

DxSin^-1(x)= 1/√(1-x^2)

22
Q

DxCos^-1(x)=?

A

DxCos^-1(x)= -1/√(1-x^2)

23
Q

DxSec^-1(x)=?

A

DxSec^-1(x)= 1/x√(x^2-1)

24
Q

DxCsc^-1(x)=?

A

DxCsc^-1(x)= -1/x√(x^2-1)

25
Q

DxCot^-1(x)=?

A

DxCot^-1(x)= -1/(x^2+1)

26
Q

DxTan^-1(x)=?

A

DxTan^-1(x)=1/(x^2+1)

27
Q

What is the Derivation of Log sub a of x? DxLogaX=?

A

The Derivation of the Log sub a of x is 1/xlna. DxLogaX= 1/((x)lna)

28
Q

What is the Derivation of the Natural log of x? DxLnx=?

A

The Derivation of the Natural log of x is 1/x. Dxlnx=1/x

29
Q

What is the Derivation of Log sub a of f(x)? DxLogaF(x)=?

A

The Derivation of Log sub a of f(x) is 1/(f(x) lna)fprime(x). DxlogaF(x)=1/(f(x)lna)fprime(x)

30
Q

What is the Derivation of the Natural log of f(x)? DxLnf(x)=?

A

The Derivation of the Natural log of f(x) is 1/f(x) *fprime(x). DxLnf(x)=1/f(x) *fprime(x)

31
Q

What is Logarithmic Differentiation? How do you use it? When do you use it?

A

Logarithmic Differentiation is using the properties of logarithms and Implicit Differentiation to find dy/dx. 1st take the ln of both sides, then simplify the Right Hand Side, then use Implicit Differentiation. If y is on the RHS, substitute the given y for it.
You use it when there is a function x to the power of another function x.

32
Q

What is the equation for the area of the circle?

A

A=pi(r)^2

33
Q

How do you evaluate a limit at infinity? (Chapter 2 Review)

A

You multiply both the numerator and the denominator by 1/x^n if n is the highest power of x found in the denominator

34
Q

What is the equation for the volume of a sphere?

A

V=4/3pi(r)^3

35
Q

How do you calculate Average Rate of Change? (Chapter 2 Review)

A

Its just the slope formula ya dummy

36
Q

What is the equation for the surface area of a sphere?

A

4pi(r)^2

37
Q

What is the equation for the volume of a cone?

A

(pi(r)^2)*(h/3)

38
Q

How do you find theta using trig?

A

Soh-Cah-Toa, (if opposite side is 4 and adjacent side is 3, sin(theta)=4/3)

39
Q

What is the derivative of n^x?

A

Dx n^x=N^xlnxfprime(x)

40
Q

What is the Law of Cosines? (For not a right triangle)

A

c^2=a^2+b^2-2(a)(b)(CosC)

41
Q

What is the Law of Sines? (For not a right triangle)

A

SinA/a=SinB/b=SinC/c

42
Q

What is the definition of the “Normal Line” in relation to the tangent line?

A

The Normal Line is the line that is perpendicular to the tangent line at the point of tangency

43
Q

What is the equation for the area of a triangle?

A

1/2base*height

Calc 140 (4 decks)

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