Implicitly Differentiating the World

1. Four types of graphs/functions that do not have a derivative are the horizontal line, the vertical line, the absolute value function, and the cusp graph.

Absolute Value

Vertical Line



2. During the process of implicit differentiation you are trying to find Dy/Dx. To do this you take the derivative of the function and using the necessary method and for each variable you take the derivative of you multiply by either Dx/Dx or Dy/Dx, depending of the variable. Then you get all the Dy/Dx on the same side then solve for Dy/Dx.

3. The most important thing to remember in a related rate problem is to always multiply by the proper term when you take the derivative of a variable.

Blog post 5

1. f' = 0 represents a possible maximum or minimum. This is because f'=0 is when the slope of f equals 0, so the function changes direction.
2. You can identify where a function increases and decreases by finding the derivative. When you find the derivative, you can plug in a value for x, in order to find the slope of the function at that point.

3. The Chain Rule is a method of find the derivative of a composite function. you begin by working from the outside of the function in. First, you take the derivative of the outside function. Second, you rewrite the interior function. Finally, you multiply by the derivative of the interior function.

 f'(x) = sqrt(1 -2x)
f'(x) = (1 - 2x)^(1/2)
f'(x) = 1/2(1 - 2x) ^ (-1/2) (-2)
f'(x) = -2/2(1 - 2x)^(1/2)
f'(x) = -1/sqrt(1 - 2x)

Find tangent line at x = -7.5
f(-7.5) = sqrt(1 -2(-7.5)) = sqrt(16)
f(-7.5) = 4

f'(-7.5) = -1 / sqrt ( 1 -2(-7.5))
f'(-7.5) = -1/4

tangent line = y - 4 = -1/4 (x + 7.5)

4. h(x) = f(g(x)) g(-4) = 5 g'(-4) =2 f'(5) = 20. Find h'(-4)
h'(-4) = f'(g(-4))(2)
h'(-4) = f'(5)(2)
h'(-4) =(20)(2)
h'(-4) = 40

Blog Post 4

Continuity

A function is continuous if…

1. The limit exists at x=a

2. f(a) exists. In other words, if there is no hole/asymptote

3. The limit at x=a is equivalent to f(a)

Example where function is not continuous:

x+3, x

f(x)= 7, x=2

x^2+3x +1, x>4

This function is not continuous because limit as x approaches 4 from the left does not equal the limit as x approaches 4 from the right.

Intermediate Value Theorem

Working example:

F(x)=2x^2-16 on interval [1,3]

F(1)=2(1)^2-16 →2-16 → f(x)=-14

F(3)= 2(3)^2-16 = 18-16 = 2

Since f is continuous on [1,3] and f(1)< 0 < f(3), then there exists c in [1,3] such that f(c) = 0

failing example:

f(x)=x2+ x -9 on interval [1,2]

f(1)=(1)2 + 1 -9 = -7

f(2) = (2)^2 + 2 -9 = -3

Since f is continuous on [1,2] and f(1)< 0 >f(4), then it cannot be concluded that there exists c in [1,2] such that f(c) = 0

Derivatives

• Type 1: The derivative that is found using the limit as h approaches 0 of the difference quotient. f(x+h) - f(x)/h.

1. f(x) = 3x + 2
2. 3(x+h) +2 - (3x + 2)/h
3. 3x+3h + 2 - 3x - 2/h
4. 3h/h
5. 3

• Type 2: The derivative that is found using the limit as x approaches a of the slope formula. f(x) - f(a)/ x-a

1. f(x)= 4x- as x approaches 1
2. 4x-4 - (4(1) -4)/x-1
3. (4)(x-1)/x-1
4. 4

A.) A derivative is a function that finds the slope of a curve of a polynomial.

B.) for me, the hardest part of finding a derivative is remembering to distribute the negative.

Instantaneous Velocity vs. Average Velocity

Instantaneous velocity is the tangent slope and one point while the average velocity is the slope of a function over an interval.

1st Calculus Blog Post

1. The most difficult topic I have encountered so hard was logarithmic functions because I was only exposed to them for a small amount of time.
2. The topic that has been the most fun in math for me was pretty much every topic because i love math.  
3. I want to live.
4. I hope to be accepted into MIT or the Naval Academy and attended there and obtain a degree in either bio-medical engineering or Nuclear engineering. I can control this by keeping my grades up, volunteering, and working hard.
5. An out of school goal of mine is to win the state championship in soccer this year. 3 things I can do to make sure this happens is work hard, lift, and focus on each game.
6. To me a function is a set of ordered pairs that do not have a repeating x-value. the function I created was y = cos(x) +  4