Category Archives: Calculus

Stewart – Calculus – 3.5 – Implicit Differentiation and Derivatives of Inverse Trigonometric Functions

httpv://www.youtube.com/watch?v=5yTVUZCaU6k


httpv://www.youtube.com/watch?v=2dv_PfEFZXY


httpv://www.youtube.com/watch?v=ximF06lmqPM


Find y'' by implicit differentiation:
9x^{2}+y^{2}=9.

18x+2yy'=0
2yy'=-18x
y'=-9x/y

y''=-9(\frac{y\cdot 1\text{--}x\cdot y'}{y^{2}})

y''=-9(\frac{y\cdot 1\text{--}x(-9x/y)}{y^{2}}) (y' is replaced with -9x/y)

y''=-9(\frac{y^{2}+9x^{2}}{y^{3}}) (after multiplying by y to eliminate denominator from -9x)

y''=-9(\frac{9}{y^{3}}) (notice that y^{2}+9x^{2} equals the original equation)

So, y''=\frac{-81}{y^{3}}.


Derivatives of Inverse Trigonometric Functions

httpv://www.youtube.com/watch?v=xBI8hVBB0pQ


Derivatives of Inverse Trigonometric Functions

Stewart – Calculus – 3.3 – Derivatives of Trigonometric Functions

\frac{d}{\mathit{dx}}(\sin x)=? \frac{d}{\mathit{dx}}(\cos x)=? \frac{d}{\mathit{dx}}(\tan x)=? \frac{d}{\mathit{dx}}(\mathrm{csc} x)=? \frac{d}{\mathit{dx}}(\mathrm{sec} x)=? \frac{d}{\mathit{dx}}(\mathrm{cot} x)=?
THE TRICK
 
To learn the derivatives of the 6 trig functions  you actually only have to learn 3 of them. The three to learn are sine, tangent, and secant.

IF
THEN
F(X) = sin X
F'(X) = cos X
F(X) = tan X
F'(X) = sec2 X
F(X) = sec X
F'(X) = sec X tan X

 
The other three functions all start with "co": 
Cosine, Cotangent, and Cosecant. 
Think of the functions as having partners:

sine
to
cosine
tangent
to
cotangent
secant
to
cosecant

 
To find the derivative of the "co" functions, start with the derivatives of sine, tangent and secant, change each function in the derivative to it’s co-function partner and put a minus sign in front of the derivative.
So starting with one of the three we need to know:

If F(x) = secant X 
then F'(x) = secant X tangent X

 So now to get the derivative of the cosecant:
 If F(X) = cosecant X 
then F'(X) = -cosecant X cotangent X

 
So all 6 trig function derivatives look like this:

IF:
THEN:
F(X) = sinX
F'(X) = cosX
F(X) = cosX
F'(X) = – sinX
F(X) = tanX
F'(X) = sec2 X
F(X) = cotX
F'(X) = – csc2 X
F(X) = secX
F'(X) = sec X tan X
F(X) = cscX
F'(X) = -csc X cot X

Thanks to Bruce Kirkpatrick for this.

Stewart – Calculus – 3.2 – Differentiation Rules

httpv://www.youtube.com/watch?v=4N4qW67m4qo


THE PRODUCT RULE

\frac{d}{\mathit{dx}}[f(x)g(x)]=f(x)\frac{d}{\mathit{dx}}[g(x)]+g(x)\frac{d}{\mathit{dx}}[f(x)]

(a) If f(x)=xe^{x}, find f'(x).

(b) Find the nth derivative, f^{(n)}(x).


Differentiate the function f(t)=\sqrt{t}(a+\mathit{bt})


If f(x)=\sqrt{x} \hspace{1 mm} g(x), where g(4)=2 and g'(4)=3, find f'(4).


THE QUOTIENT RULE

\frac{d}{\mathit{dx}}[\frac{f(x)}{g(x)}]=\frac{g(x)\frac{d}{\mathit{dx}}[f(x)]\text{--}f(x)\frac{d}{\mathit{dx}}[g(x)]}{[g(x)]^{2}}

Let y=\frac{x^{2}+x\text{--}2}{x^{3}+6}


Find an equation of the tangent line to the curve y=\frac{e^{x}}{(1+x^{2})} at the point (1,\frac{1}{2}e).


 \frac{d}{\mathit{dx}}(c)=?   \frac{d}{\mathit{dx}}(x^{n})=?   \frac{d}{\mathit{dx}}(e^{x})=?   (\mathit{cf})'=?   (f+g)'=?   (f-g)'=?   (\mathit{fg})'=?   (\frac{f}{g})'=?
SOLUTION 1
Using the Product Rule
 

SOLUTION 2
If we first use the laws of exponents to rewrite f(t), then we can proceed directly without using the Product Rule.

0

Stewart – Calculus – 2.5 – Continuity

  1. Definition: A function f is continuous at a number a if
     

    This definition implicitly requires three things if f is continuous at a:

    1. f(a) is defined (that is, a is in the domain of f)

    2. \lim_{x \to a}f(x) exists

    3. \lim_{x \to a}f(x)=f(a)


  2. If f and g are continuous functions with f(3)=5 and
    \lim_{x \to 3}[2f(x) - g(x)]=4, find g(3).


    Use the definition of continuity and the properties of limits to show that the funtion is continous at the given number a. f(x)=(x+2x^{3})^{4}, a=-1


  3. The Intermediate Value Theorem
    Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where f(a) \neq f(b). Then there exists a number c in (a,b) such that f(c)=N.

    ~ ~ ~

    Use the Intermediate Value Theorem to show that there is
    a root of the given equation in the specified interval.

    x^4+x-3=0, (1,2)

Calculus Test 1

  1. Specify the domain of the function y = |x – 1|.

  2. Which of the following is a true statement about the graph of the equation y = x^4 +1?
  3. 1. It is symmetric about the x-axis.
    2. It is symmetric about the y-axis.
    3. It has two x-intercepts.
    4. It has no x-intercepts.

    a. Only statements 2 and 3 are true.
    b. Only statements 2 and 4 are true.
    c. Only statements 1 and 3 are true.
    d. Only statement 2 is true.


  4. Which of the following statements are true of the graph of y = \frac{2x-1}{x+1} ?

    1. It has no x-intercept.
    2. It has a slant asymptote y=2x.
    3. It has a vertical asymptote at x=-1.
    4. It has a horizontal asymptote at y=2.

    a. Statements 1, 2, and 3 are true.
    b. Statements 3 and 4 are true.
    c. Statements 2, 3, and 4 are true.
    d. All four statements are true.


  5. Let f(x) = 2 \cos x. The domain of f^{-1}(x) is

    1. [-1,1] 2. (2,  \infty) 3. (-\infty,  \infty) 4. (-2,2) 5. [-2,2]


  6. Simplify as far as possible. \ln e +a^{\log_{a}5}- \log100+10^0 - \log_{3} \frac{1}{3}
  7. Solve for x.

    a. \log_3 x + \log_3(2x+5)=1

    b. \frac{1}{\sqrt{2}}=4^{x}


  8. f(x)=(x-3)(x+1)^2(x-1)^4
  9. Identify the parts of the following composite function. f(g(h(j(x))))=\frac{1}{\sqrt{\log(x-1)}}
  10. Consider the picture below. Find the equation of the line L.


  11. The following table represents a function of the form f(x)=ab^{x}. Find the equation of the function.

     \begin{tabular}{| l | c | r |}  \hline  x & f(x)\\ \hline    0 & 6\\ \hline    1 & 18\\ \hline    2 & 54\\ \hline    3 & 162\\ \hline    4 & 486\\  \hline  \end{tabular}
  12. Given a^{m}=2, a^{n}=3, b^{m}=4, and b^{n}=5,
    use the properties of exponentials to determine (a^{3n}b^{m+n})^{\frac{1}{3}}.


  13. Evaluate:
    1. \sin \frac{\pi}{3}
    2. \tan \frac{3\pi}{4}
    3. \cos \frac{5\pi}{6}
  14. Find all solutions to the equaion, \tan^{2}x=\tan x, such that x \in [0,2\pi].
    Express the answers in radians.


  15. Find f^{-1}(x) if f(x) =  \sqrt{e^{x}+2}.
all real numbers
b
b
5
6
1. Set x equal to 0 to find the y-intercept.
2. Set the values in parentheses equal to 0 to get the x-intercepts 3, —1, and 1.
4. Note that the function will not cross the x axis with an even power, but will cross with an odd power.