Sam's Book of Calculus

Table of Contents

1. Trig Identities                                          2. Basic Properties of Limits

3. Continuity                                               4. Limit Definition of Derivative

5. Functions with No Derivative                 6. Derivatives

7. Implicit Differentiation                            8. Relative Minimum and Maximum

9. Rolle's Theorem                                    10. 1st Derivative Test

11. 2nd Derivative Test                              12. Anti-derivatives

13. Find Area Approximations                    14. Area Under the Curve

15. Fundamental Theorems of Calculus    16. U-Substitution

1. Trig Information

Graphs

Reciprocal Properties

Sin = opp/hyp = 1/csc

Cos = adj/hyp = 1/sec

Tan = opp/adj = sin/cos= 1/cot

Csc = hyp/opp = 1/sin

Sec = hyp/adj = 1/cos

Cot = adj/opp = cos/sin = 1/tan

Simple Trig Values

Derivatives and Antiderivatives

Trig Identities

Pythagorean Identities

cos²(x) + sin²(x) = 1

1 + tan²(x) = sec²(x)

cot²(x) + 1 = csc²(x)

Addition and Subtraction Properties

Cos(a+β)=cos(a)cos(β)-sin(a)sin(β)

Sin(a+β)=sin(β)cos(a)+sin(a)cos(β)

Cos(a- β)=cos(a)cos(β)+sin(a)sin(β)

Sin(a- β)=sin(a)cos(β)-sin(β)cos(a)

Ex: cos(30+20)=cos(30)cos(20)-sin(30)sin(20)

Double Angle Properties

Sin(2x) = sin(x+x) = 2 sin(x) cos(x)

Cos(2x) = cos(x+x) = cos²(x)-sin²(x) = 1-2sin²(x) = 2 cos²(x)-1

Tan(2x) = tan(x+x) = 2 tan(x) / 1-tan²(x)

2. Basic Properties of Limits

Basic Properties

Dividing Out

If factors can cancel out, then the function is not continuous at that point.

Ex: x³-27 / x²-x-6 = (x-3)(x-3)(x-3) / (x-3)(x+2) = hole at x=3, limit DNE at x=3

Squeeze Theorem

h(x) < f(x) < g(x)

If limit as x→a for h(x) = limit as x→a for g(x), then the limit as x→a for f(x) exists

Ex: f(x) x→0 = x cos(1/x)

    = -1 < cos(1/x) < 1

    = -x < x cos(1/x) < x

Infinite Limits

As x gets nearer and nearer to infinite, the function gets closer and closer to 0.

Piecewise Limits

Ex: f(x) = { x+1    x<1

                { x-1    x>1

f(1) does exist on f(x), so there is a limit at f(1). However, f(x) is not continuous at x=1

Left Side Limits and Right Side Limits

Limits from the left and right sides can determine continuity. Both the limit from the left and the limit from the right have to equal the same value for the function to be continuous.

Ex: f(x) = { x+1    x<1

                { x-1    x>1

   1. f(x) from the left (use the first function) - lim x→1- = 2

  2. f(x) from the right (use the second function) - lim x→1+ = 0


3. Continuity

For a Function to be Continuous:

1. Limit as x approaches a from the left = limit as x approaches a from the right

2. f(a) exists

3. f(a) = limit as x approaches a

Examples:


Ex: lim x→1 x³+1

    1. f(1) = 2

    2. lim x→1- = 2, lim x→1+ = 2

    3. f(2) = lim x→2

    f(x) is continuous at x=1

Ex: lim x→2 x²-4 / x-2

    1. f(2) = Undefined

    2. lim x→2- = 4, lim x→2+ = 4

    3. f(2) Does Not Equal lim x→2

    f(x) is not continuous at x=2 (f(a) does not equal the limit as x→2)

Piecewise Function Examples:

Ex: lim x→2 f(x) = { x²+1   x<2

                             { 2x       x>2

    1. f(2) = 4

    2. lim x→2- = 5, lim x→2+ = 4

  f(x) is not continuous at x=2 (limits from left and right are not the same)

Ex: lim x→2 f(x) = { x²+1    x<2

                             { 3x-1    x>2

    1. f(2) = 5

    2. lim x→2- = 5, lim x→2+ = 5

    3. f(2) = lim x→2

    f(x) is continuous at x = 2

4. Limit Definition of Derivative

Two formulas used to find the derivative of the function the long (and hard) way.

1. lim x→c = f(x) - f(c) / x-c

    Used for functions with multiple x's (x²-x)

2. lim h→0 = f(x+h) - f(x) / h

    Can be used for all functions - works every time

5. Functions With No Derivative

Functions with a corner or cusp do not have a derivative. Basically, functions that come to a point will not have a derivative.

Corner:

Cusp:

6. Derivatives

Constant Functions

The derivative of a constant always equals zero.

Ex: f(x) = 7 , f'(x) = 0

Power Rule

To find the derivative, bring the exponent down in front and subtract 1 from the actual exponent.

Ex: f(x) = x² , f'(x) = 2x^(2-1) = f'(x) = 2x

Equations of Tangent Lines

To find the equation of the tangent line, plug the the x value of your point into the derivative of f(x). This will give you your slope (m) for the tangent line. Plug the point and slope into the point slope formula [y-y1 = m(x-x1)]

Ex: find the equation of the line tangent to f(x) = 2x² at (1,2)

    1. f'(x) = 4x

    2. 4(1) = 4 = m

    3. y-2 = 4(x-1)

    4. y = 4x-2 is the equation of the tangent line

Product Rule

Ex: d/dx (2x+1)(x²)

    1. 2(x²) + 2x(2x+1)

    2. 2x² + 4x² + 2x

    3. f' = 6x² + 2x

Quotient Rule

Ex: d/dx (x²) / (3x)

    1.{[3x (2x)] - [x² (3)]} / (3x)²

    2. f' = (6x - 3x²) / 9x²

Higher Order Derivatives

1st derivative = slope or velocity (Derivative of f(x))

2nd derivative = concavity or acceleration (Derivative of f'(x))

Ex: f(x) = x³+x²+x+2

    1. f' = 3x²+2x+1

    2. f'' = 6x+2

Chain Rule

Used for composite functions. f(x) = g(h(x)).

    1. Bring the exponent down in front and subtract one from it.

    2. Multiply by the derivative of h(x).

Ex: f(x) = (x² + 3x)³

    1. 3(x² + 3x)²

    2. f'(x) = 3(x² + 3x)² * (2x +3)

Ex: f(x) = sin²(3x)

    1. 2cos(3x)

    2. 2cos(3x) * 3

    3. f'(x) = 6cos(3x)

7. Implicit Differentiation

Used to find the derivative for f(x) when we have more than one variable (x and y).

Ex: d/dx[x+3y]

    1. d/dx[x] + d/dx[3y]

    2. 1 + 3*dy/dx

Ex: 3x² + 4y² = 7

    1. d/dx[3x²] + d/dx[4y²] = d/dx[7]

    2. 6x + 8y*dy/dx

    3. dy/dx = -3x/4y

Ex: d/dx[xy²]

    1. d/dx[x]*y² + d/dx[y²]*x

    2. 1*y² + dy/dx*2xy

    3. y² + 2xy*dy/dx

Related Rates

Implicit Differentiation in terms of t: used to find the rate of increase/decrease.

Ex: A=πr², dr/dt=1 ft/sec, @ 4 seconds (r=4)

    1. d/dt[A=πr²]

    2. dA/dt = dr/dt*2πr

    3. dA/dt = (1 ft/sec)*2π(4)

    4. dA/dt = 8π ft/sec

Ex: V=s³, ds/dt = 3cm³/sec, @ 1 and 10 seconds (r=, r=)

    1. d/dt[V=s³]

    2. dV/dt = ds/dt*3s²

    3. dV/dt = (3cm³/sec)*3(1)²            →           dV/dt = (3cm³/sec)*3(10)²

    4. dV/dt = 9cm³/sec                      →           dV/dt = 900cm³/sec

8. Relative Min and Max

Critical points will tell us where f(x) has a relative min or max or where f(x) does not exist. To find critical points, find the derivative of f(x) and solve for when f' = 0.

Ex: y = x³-3x+2

    1. f'(x) = 3x²-3

    2. 0 = 3x²-3

    3. 3 = 3x²

    4. 1 = x²

    5. x = + 1       →       (+1 are critical numbers)

Plug your critical numbers back into f'(x) along with the numbers on either side or in between them to determine whether they are relative min or max or DNE.

f'(x) -2    -1    0    1    2                                                                                                           9     0    -3    0    9    (y values are the slope of the function)

Based on the graph, -1 is a relative maximum and 1 is a relative minimum.

Ex: x³-4x+6

    1. f'(x) = x²-4

    2. x = + 2

f'(x) -3    -2    0    2    3                                                                                                           5     0    -4   0    5

Based on the chart, -2 is a relative maximum, and 2 is a relative minimum.

9. Rolle's Theorem

If f is continuous on [a,b], AND f(a) = f(b), then on (a,b) there is at least one critical point.

Ex: f(x) = x²-2x on [0,2]

    1. f(0) = 0 = f(2)                 (There is at least one critical point)

    2. f' = 2x - 2

    3. Critical point at x=1       (Plug back into original to get y value)

    4. (1,-1) is the critical point, it is a relative minimum

Ex: x²-5x+4 on [1,4]

    1. f(1) = 0 = f(4)

    2. f' = 2x-5

    3. Critical point at 5/2

10. 1st Derivative Test

Used to find whether a critical point is a relative min or max, and where the function is increasing or decreasing.

Let c be a critical number on f.

1. If f' changes from + to - at c, then f(c) is a relative maximum

2. If f' changes from - to + at c, then f(c) is a relative minimum

3. If there is no change at c, then it is not a max or a min

Ex: (x^4+1) / x²

    1. f' = (2x^4-2) / x³

    2. Critical numbers: x = -1, 0, 1

    3. f'(x) -2    -1    -1/2    0    1/2    1    2                                                                                         -      0       +     0      -      0    +

Based on the chart, -1 is a relative min, 0 is a relative max, and 1 is a relative min. Since f'(x) is the slope, the chart tells us that f(x) decreases from (-,-1) and (0,1), and increases from (-1,0) and (1,)

11. 2nd Derivative Test

Used to find the inflection points (where f" = 0) which tell us where the curve changes concavity on a function.

Let f''(c) = 0 and f'' exists on an open interval around c

1. if f''(c) > 0, then (c, f(c)) is a relative minimum and is concave up.

2. if f''(c) < 0, then (c, f(c)) is a relative maximum and is concave down.

Ex: x³-6x²+12x

    1. f' = 3x²-12x+12

    2. Critical points: x = 2

        f'(x) = 1    2    3                                                                                                                         3    0    3

        2 is a relative minimum

    3. f'' = 6x - 12

    4. Inflection points: x = 2

        f'(x) = 1    2    3                                                                                                                         -6    0    6

        f(x) is concave down from (-∞,2), f(x) is concave up from (2,∞)

12. Antiderivatives

General Antiderivatives

Use the Reverse Power Rule to find the antiderivative of a function. Don't forget to add c!

Ex: f'(x) = x²-4 dx

    1. f(x) = 1/3x³-4x+c

Ex: f'(x) = 2x-3 dx

    1. f(x) = x^2-3x+c

Ex: f'(x) = -cos(x)+2

    1. f(x) = -sin(x)+2x+c

Finding C

To solve for c, you must be given f(x) = n. Plug x into f(x) and set it equal to n.

Ex: f'(x) = -2x+3     f(0) = 0

    1. f(x) = -x^2+3x+c

    2. 0 = -(0)^2+3(0)+c

    3. c= 0

    4. f(x) = -x^2+3x

13. Find Area Approximations

Use summations to find area approximations. Σ means the summation, change in x is the interval ( [1, 4], 4-1 = 3) over the desired number of rectangles.

14. Area Under the Curve

Can be found using limit definitions and summations, or definite integrals. Using left or right endpoints in a summation will give you an area approximation, while using the definite integral or the limit definition and summation will give you the exact area under the curve.

Definite Integral:

15. Fundamental Theorem of Calculus

FTC Part 1

FTC Part 2

16. U Substitution

Used for antiderivatives with complicated variables, such as the antiderivative of         x(x²+1)³dx, or the antiderivative of sin²(3x)cos(3x)dx.

1. Pull out the factor to sub u in for.

2. Take the derivative of u and multiply by dx to get du

3. Plug u in for factor, and multiply by the coefficient and reciprocal of the coefficient of du

4. Simplify

5. Plug factor back in for u

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