Why study calculus?
Calculus is the mathematical analysis of functions using limit approximations. It deals with rates of change, motion, areas and volumes of regions, and the approximation of functions. Modern calculus was developed in the middle of the 17th century by Isaac Newton and Gottfried Leibniz. Today, calculus is the essential language of science and engineering, providing the means by which real-world problems are expressed in mathematical terms.
Example (Newton)
You drop a rock from the top of a cliff at Lake Powell that is 220 feet above the water. How fast is the rock traveling when it hits the water?
Example (Leibniz)
How can you determine the maximum and minimum values of a function? On what intervals is it increasing? On what intervals is it decreasing?
Suppose y = f (x) = x2 +1 .
a) Sketch the graph of f . b) Plot the points (1, f (1)) and (1+ h, f (1+h )). c) Draw the line containing these two points. d) Find the slope of this line (secant line). e) What happens to the slope when h is closer and closer to zero? f) Find the equation of the line tangent to the graph of f at the point (1,2 ).
Definition of the Limit
lim f (x) L means x→c =
For every distance ε > 0 there exists a distance δ > 0 such that if 0 < x−c <δ then 0 < f (x)− L <ε .
How can the limit of a function fail to exist as x approaches a value a ?
Consider the following three functions:
x a) f where f (x) = x
b) g where g(x)= 1 x2
1 c) h where h(x)= sin x
One-Sided Limits
What can you say about the function f whose graph is given below?
Theorem
lim f (x) = L if and only if lim f (x) = L and lim f (x) = L . x→c x→c− x→c+
sin Example: Show that lim θ = 1 θ→0 θ
Continuous Functions
Definitions:
Suppose f is defined on the interval [ a , b ] and a < c < b .
If lim f (x) = f (c) then f is continuous at c . x→c
If lim f (x) = f (a) then f is right-continuous at a . x→a+
If lim f (x) = f (b) then f is left-continuous at c . x→b+
If f is right-continuous at a , left-continuous at b , and continuous at each point c in ( a , b ) then f is continuous on the interval [ a , b ] .
You can draw the graph of f without lifting your pen.
Properties of continuous functions:
Intermediate Value Theorem
If f is continuous on [ a , b ] and m is any value between f (a) and f (b) then there is a least one value c in [ a , b ] such that f (c) = m .
Limits at Infinity → ∞
1. lim f (x) L means for each 0 there exists M > 0 such x→∞ = ε > that for each x , if x > M then f (x)− L < ε .
2. lim f (x) means for each B > 0 there exists 0 such x→c = ∞ δ > that for each x , if 0 < x − c <δ then f (x) > B .
3. lim f (x) means for each B > 0 there exists M > 0 x→∞ = ∞ such that for each x , if x > M then f (x) > B .
Definition of the Derivative
The derivative of a function f at a point x0 , denoted f '(x0) , is
f (x + h)− f (x ) f '(x )= lim 0 0 provided this limit exists. 0 h→0 h
1. f '(x0) is the slope of the line tangent to the graph of f at
the point ( x0, f (x0)) .
2. f '(x0) is the instantaneous rate of change of f at x = x0 .
Note that
f (x0 + h)− f (x0) f (x)− f (x0) f '(x0)= lim = lim h→0 x→x0 h x − x0
One- sided derivatives:
Suppose the function f is define on [ a,b ] .
If lim f (a+ h)− f (a) then f is said to have a right-derivative h→0+ h or derivative from the right at x = a .
If lim f (b+ h)− f (b) then f is said to have a left-derivative h→0− h or derivative from the left at x = b .
f is differentiable on [ a,b ] provided it is differentiable on ( a,b ), has a derivative from the right at x = a and a derivative from the left at x = b .
dy df Notation: y = f (x) , y' , dy /dx , , dx dx
Theorem. If f is differentiable at x = x0 then f is continuous at x = x0 . Differentiation Rules
1. Power Rule
d xn = nxn−1 dx
2. If f (x)= c for each x then f '(x)= 0 for each x .
The derivative of a constant function is 0 .
3. f (x)= c⋅u(x) then f '(x)= c⋅u'(x) .
The derivative of a constant times a function is the constant times the derivative of the function.
4. Sum Rule
Suppose w(x)= f (x)+ g(x) then w'(x)= f '(x)+ g'(x) .
The derivative of the sum of two functions is the sum of the derivatives.
5. Product Rule
Suppose w(x) = f (x)⋅g(x) then w'(x) = f (x)⋅g'(x)+ g(x)⋅ f '(x) .
The derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.
f (x) g(x)⋅ f '(x)− f (x)⋅g'(x) If w(x) = then w'(x) = 2 . g(x) ⎡ ⎤ ⎣g(x)⎦ The derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all over the denominator squared.
7. Exponential Functions
If y = ex then y' = ex .
If y = ax then y' = ax lnx .
1 If y = lnx then y' = for x > 0 . x d 1 log x = for x > 0 dx a xlna
8. Higher-Order Derivatives
y' , y" , f "(c) , d2y /dx2, f (3)(x) , f (n)(x)
Derivatives of Trig Functions
d d sinx = cosx cosx = −sinx dx dx
d d tanx = sec2 x secx = secxtanx dx dx
d d cot x = − csc2 x cscx = − cscx cot x dx dx Chain Rule
If w = go f and w(x) = g ⎡ f (x) ⎤ then w'(x) = g'⎡ f (x) ⎤ f '(x) . ⎣ ⎦ ⎣ ⎦
The derivative of the composite of two functions is the derivative of the outside function (evaluated at the inside function) times the derivative of the inside function.
Implicit Differentiation
Method of Implicit Differentiation
1. Given an equation in x and y , differentiate both sides with respect to x under the assumption that y is a differentiable function of x.
2. Solve the resulting equation for dy . dx
Inverse Functions
Suppose f :A → B & f is 1−1 onto B . Then f −1 :B → A is defined by f −1(y) = the value x in A such that f (x)= y.
Suppose y = f (x) , f is 1−1, and f ⎡ f −1(x)⎤ = x . Using the ⎣⎢ ⎦⎥ chain rule we get
d f '⎡ f −1(x)⎤ ⋅ f −1(x) = 1 and ⎣⎢ ⎦⎥ dx
d 1 f −1(x) = . dx f '⎡ f −1(x)⎤ ⎣⎢ ⎦⎥
Inverse Trigonometric Functions
Applications of Mathematics – Problem Solving Steps
1. Understand what is given. 2. Choose a variable (s). 3. Obtain a mathematical description of the problem. 4. Do the math. 5. Interpret the results.
Related Rates
Describe the applied problem mathematically in terms of the related variables, then differentiate implicitly.