Flashcards in Math 150 Chapter 2.7 - 3.10 Deck (69)

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1

## What is the definition of the tangent line?

###
The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope

m = lim┬(x→a) [f(x)-f(a)]/(x-a) provided that this limit exists

2

## What is the definition of derivative?

### The derivative of a function f at a number a, denoted by f'(a) is f'(a)= lim┬(h→0) [f(a+h)-f(a)]/h if this limits exists

3

## What is the definition of instantaneous velocity?

###
We define the velocity (or instantaneous velocity) v(a) at time t = a as

v(a) = lim┬(h→0) [f(a+h)-f(a)]/h

4

## Define rates of change and the average rate of change

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Let f be a function defined on an interval I and let

x1, x2 ∈ I. Then the increment of x is defined as

∆x = x2 − x1

and the corresponding change in y is

∆y = f(x2) − f(x1)

The average rate of change of y with respect to x over the interval [x1, x2] is defined as

∆y/∆x= [f(x subscript:2) − f(x subscript:1)]/(x subscript: 2 − x subscript: 1)

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## What is the definition of instantaneous rate of change of y with respect to x?

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The instantaneous rate of change of y with respect to x is defined as lim┬(∆x→0) (∆y/∆x) =

lim┬(x subscript:2→ x subscript:1) [f(x subscript:2) − f(x subscript:1)]/(x subscript: 2 − x subscript: 1)

6

## What does it mean if f is a function and f': J→R?

### If a function f : I → R is given, find the set J ⊂ I such that f'(x) exists for each x ∈ J. If J ≠ ∅ then this new function f': J → R is called the derivative of f.

7

## What is the definition of differentiable?

###
A function is differentiable at a if f'(a) exists. It is differentiable on an open

interval (a, b) [or (a, ∞) or (−∞, a) or (−∞, ∞)] if it is differentiable at every number in the interval

8

## Prove that every differentiable function is continuous

###
Consider a differentiable function f at x=a. We want to show that lim┬(x→a) f(x) = f(a). This is the continuous definition. Bring one side to the other so that the function f equals zero. [lim┬(x→a) f(x)] - f(a) = 0. Because f(x) is continuous on f(a), we can apply the limit laws on f(a) so that lim┬(x→a)〖f(x) - f(a)〗=0 and then multiply the expression by (x-a)/(x-a).

lim┬(x→a)〖(f(x) - f(a))/(x-a)〗multiplied by lim┬(x→a) (x-a). By DSP, the equation would equal zero and therefore every differentiable function is continuous by the definition of continuity.

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## Is every continuous function differentiable? Prove your answer. (3 things)

###
No. Three counterexamples would be:

- When the graph of f has a corner/cusp at the point (a, f(a))

- f is not continuous at a

- The graph of f has a vertical tangent line when x=a

10

## What is the definition of vertical tangent line?

### If a function is continuous as x=a and the limit as x approaches a is infinite or negative infinity, then the function has a vertical tangent line at x=a

11

## What is d/dx of (sin x)?

### cos x

12

## What is d/dx of cos x?

### -sin x

13

## What is d/dx of tan x?

### sec^2 x

14

## What is d/dx of sec x

### (sec x)(tan x)

15

## What is d/dx of csc x?

### -(csc x)(cot x)

16

## What is d/dx of cot x?

### -(csc^2 x)

17

## What are the three Pythagorean identities?

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sin^2 (t) + cos^2 (t) = 1

tan^2 (t) + 1 = sec^2 (t)

1 + cot^2 (t) = csc^2 (t)

18

## What are the other three trig identities?

###
sin(–t) = –sin(t)

cos(–t) = cos(t)

tan(–t) = –tan(t)

19

## What are six Angle-Sum and -Difference Identities?

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sin(α + β) = sin(α) cos(β) + cos(α) sin(β)

sin(α – β) = sin(α) cos(β) – cos(α) sin(β)

cos(α + β) = cos(α) cos(β) – sin(α) sin(β)

cos(α – β) = cos(α) cos(β) + sin(α) sin(β)

tan(α+β) = [tan(α)+tan(β)] / [1-tan(α)tan(β)]

tan(α−β) = [tan(α)−tan(β)] / [1+tan(α)tan(β)]

20

## What are three double-angle identities?

###
sin(2x) = 2 sin(x) cos(x)

cos(2x) = cos^2 (x) – sin^ 2(x) = 1 – 2 sin^ 2 (x) =

2 cos^2 (x) – 1

tan(2x) = [2tan(x)] / [1−tan^2 (x)]

21

## What is the sine law?

### sin A/a = sin B/b

22

## What is the cosine law?

### c^2=a^2+b^2-2abcos(C)

23

## What is the equation of a tangent line?

### Suppose y= f(x) at (a,f(a)). Then the equation of the tangent line is y-f(a)=f'(a)(x-a)

24

## What is the power rule?

### If n is any real number, then [d/dx] x^n = nx^(n-1)

25

## Prove the power rule: [d/dx] x^n = nx^(n-1)

###
For some integer n, [d/dx] x^n =lim┬(h→0)〖[(x+h)^n - x^n] / h 〗= lim┬(h→0) [x^n+nhx^(n-1)+......+h^n - x^n] / h

= nx^(n-1) = f'(x)

26

## What is the constant multiple rule and prove it

### If c is a constant and f is a differentiable function, then [d/dx] (cf(x)) = c [d/dx] f(x). Proof: [d/dx] (cf(x)) = lim┬(h→0) [c(f(x+h) - c(f(x))} / h = c( lim┬(h→0) [(f(x+h) - (f(x))} / h) = c [d/dx] f(x)

27

## What is the sum rule?

### If f and g are differentiable functions, then (d/dx) (f(x)+g(x)) = (d/dx) (f(x)) + (d/dx) g(x)

28

## Prove the sum rule: (d/dx) (f(x)+g(x)) = (d/dx) (f(x)) + (d/dx) g(x)

### (d/dx) (f(x)+g(x)) = lim┬(h→0) [f(x+h)+g(x+h)-f(x)-g(x)] / h = lim┬(h→0) [f(x+h)-f(x)+g(x+h)-g(x)] / h = lim┬(h→0) [f(x+h)-f(x)] / h + lim┬(h→0) [g(x+h)-g(x)] / h = f'(x) + g'(x)

29

## If f(x) = a^x, a>0, a≠1, is an exponential function then ....

### If f(x) = a^x, a>0, a≠1, is an exponential function then f'(0)=lim┬(h→0) [a^h - 1] / h exists.

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