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Flashcards in Chapter 3 Deck (43)
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1

What is the constant rule? (DxC=?)

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

2

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

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

3

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

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

4

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

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

5

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

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

6

What is the definition of the number e?

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)

7

What is the product rule? ((f*g*l*m)prime)=?

((f*g*l*m)prime= fprime*g*l*m+f*gprime*l*m+f*g*lprime*m+f*g*l*mprime

8

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

(f/g)prime=(fprime*g-f*gprime)/g^2

9

DxSinx=?

Cosx

10

DxCosx=?

-Sinx

11

DxTanx=?

Sec^2x

12

DxCscx=?

-CscxCotx

13

DxSecx=?

SecxTanx

14

DxCotx=?

-Csc^2x

15

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

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

16

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

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

17

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

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)

18

What is the derivative of e^x?

Dx e^x=e^x

19

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

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

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

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

DxSin^-1(x)=?

DxSin^-1(x)= 1/√(1-x^2)

22

DxCos^-1(x)=?

DxCos^-1(x)= -1/√(1-x^2)

23

DxSec^-1(x)=?

DxSec^-1(x)= 1/x√(x^2-1)

24

DxCsc^-1(x)=?

DxCsc^-1(x)= -1/x√(x^2-1)

25

DxCot^-1(x)=?

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

26

DxTan^-1(x)=?

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

27

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

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

28

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

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

29

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

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

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

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

31

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

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

What is the equation for the area of the circle?

A=pi(r)^2

33

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

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

What is the equation for the volume of a sphere?

V=4/3pi(r)^3

35

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

Its just the slope formula ya dummy

36

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

4pi(r)^2

37

What is the equation for the volume of a cone?

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

38

How do you find theta using trig?

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

39

What is the derivative of n^x?

Dx n^x=N^x*lnx*fprime(x)

40

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

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

41

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

SinA/a=SinB/b=SinC/c

42

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

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

43

What is the equation for the area of a triangle?

1/2base*height