CALC 1 CLEP

Limit

lim x -> a- f(x) = lim x -> a+ f(x)

Direct Sub. Law

lim x -> a f(x) = f(a)

*a is in domain of f

lim x -> a f(x) = f(a)

Limit Definition of Derivative

f'(x) = lim h-> 0 f(x+h)+ f(x) / h

Discontinuities

Jump, Infinite, removable

-Jump: holes, limits DNE

-Removable: holes, limits exists

-Infinite

Relationship between Differentiability and Continuity

Differentiability Implies Continuity and

Continuity does not imply Differentiability

Continuity Implies Integrability

Unit Circle

Trigonometry Table

Angle Sum Identities

sin(A+B) = sin(A)cos(B) + cos(A)sin(B)term-8

cos(A+B) = cos(A)cos(B) - sin(A)sin(B)

Relationship w/ Applications of Derivatives on Position, Velocity and Acceleration

Position -> Velocity -> Acceleration : Derivative

Acceleration -> Velocity -> Position : Integral

If fun. fails to be differentiable, then it is either cases...

-Discontinuities

-Kink/Corner/Cusp

-Vertical Tangent Line

Continuous

-f is continuous at x if lim y->x f(y) = f(x)

-f is continuous if it is continuous at x for all x

-Polynomials, exponential fun., sin, cos are all continuous

-Rational functions

-Log is continuous

Squeeze Theorem

When direct substitution fails, h(x)

Differentiability

Smoothness of curve

Standard Differentiation/Derivatives Rules

-Derivative of power fun. = (x^n)' = nx^n-1

-Derivative of constant = f'(x) = 0

-Derivative of sum/subtract rule = f'(x) + g'(x)

-Derivative of constant multiple = c ⋅ f'(x)

Tangent Line Slope & Equation

Slope of tangent line(*use derivative or derivative of limit rule):

f'( given x ) = m

then point slope form with given x and y:

y - y1 = m(x -x1)

******************to find y1, f(given x)

Average rate of change of f on (a, b)

(f(b)-f(a)) / b-a

Instantaneous change of f

f'(x)

Product Rule - Derivative

f'(x)g(x)+f(x)g'(x)

Quotient Rule Of Derivatives

f'(x)g(x)-f(x)g'(x) / g(x)^2

Chain Rule - Derivative

f(g(x))' = f'(g(x)) ⋅ g′(x)

[f(x)]^n = n[f(x)]^n-1 ⋅ f'(x)

Trigonometry Derivatives

d/dx [sin(x)] = cos(x)

d/dx [cos(x)] = -sin(x)

d/dx [tan(x)] = sec(x)^2

d/dx [cot(x)] = -csc(x)2

d/dx [sec(x)] = sec(x)tan(x)

d/dx [csc(x)] = -csc(x)cot(x)

d/dx [arcsin(x)] = 1/√(1-x^2)

d/dx [arctan(x)] = 1/(1+x^2)

Exponential and Logarithmic Derivatives

[ex]' = ex

[nx]' = nx · ln(n)

[ln(x)]' = 1/x

[logn(x)]' = 1/[ln(n)x]

Pythagorean Identity

cos^2x + sin^2x=1

Trigonometric Reciprocal Identites

sin(x) = opp/hyp

cos(x) = adj/hyp

tan(x) = opp/adj

csc(x) = 1 / sin

sec(x) = 1 / cos

cot(x) = 1 / tan

Higher Order Derivatives

First : f'(x)

Second : f''(x)

Third : f'''(x)

nth : f^(n) (x)

Implicit Differentiation

y' ⋅ dy/dx then solve for dy/dx

L'Hospital's Rule

If the limit of a function is indeterminate (i.e. 0/0 or ∞/∞) then you can take the derivatives of the top and bottom functions separately, provided the second limit exists

*do it again if indeterminate form again, continuing taking derivative till answer is not indeterminate

Rolle's Theorem

If f(x) is continuous and differentiable on [(a, b)] and f(a) = f(b), then there is at least one value of x on (a, b), call it c, such that f'(c) = 0

Inverse Function - Derivative

(f^-1(a))' = 1/f'(b)

Increasing Interval

f'(x0) > 0 ; positive

Decreasing Interval

f'(x0) < 0 ; negative

Critical Points

f'(x0) = 0 or undefined (First Derivative Test)

Concave up

f''(x) > 0

Concave down

f''(x) < 0

Inflection Points

f''(x) = 0 or undefined

points where concavity of f"(x) changes

Local Min / Relative Minima

f''(x) > 0 (pos.) and f''(x) = 0

from - to +

Local Max / Relative Extrema

f''(x) < 0 (neg.) and f''(x) = 0

from + to -

Mean Value Theorem

f'(c) = (f(b) - f(a))/ (b - a)

*Conditions met : continuous on interval [a,b] and differentiable on interval [a,b]

Indefinite Integral

∫ f(x) dx = F(x) + C

Definite Integral

a->b ∫ f(x) dx = F(b) - F(a)

Integral/Antiderivative Rules

-power rule = x^n+1 / n+1

*if n ≠ -1

-if n = -1, ∫ x^n dx = ln(x) + c

-same sum and constant rule as regular derivative rules (i.e. take constant next to variable out of integral, etc.)

***but if constant alone (i.e. 1), x^0+1 / 0+1

U-Substitution (or Reverse Chain Rule)

∫ f(x)*g(x) = ∫ u * du OR

∫ f(g(x)) dx = ∫ f(u) du

Where u = g(x) and du = g'(x) dx (*DU CANCELS OUT)

Ex.

-∫ cos (ax) = 1/ a sin(ax) + c

-∫ e^ax = 1/a e^ax + c

-∫ ln(ax) = 1/a ln∣ax∣ + c

Riemann Sum

^n ∑ i=1 f(xi∗)Δx

-n=number of subintervals

-i = starting index of the summation

-f(xi∗) is the value of the function at a sample point within the 𝑖ith subinterval.

-Δx is the width of each subinterval.

ex.

^50 ∑ i=1 (i/50)^2 ⋅ 1/50 = integral 0-1 x^2 dx

represents the sum of areas of 50 rectangles, each with a width of 1/50 and a height given by the function f(x) = x^2 at the right endpoint of each subinterval i/50, where i ranges from 1 to 50.

Finding Left, Right Riemann Sum

Given function f on [a,b] and n subintervals of equal length (x=[a to b]=width),

L : (left most x values = n width, corresponding y-values = length) - Total Area += width*length for every subinterval

R : (find using same format as left except right most x= n width)

|L-R| = Total Area(L) - Total Area(R)

Fundamental Theorem of Calculus PT1

- d/dx ∫ f(x)dx = f(x)

*Derivative and integral undo each other

Fundamental Theorem of Calculus PT2

- ∫ f(x) dx on interval a to b = F(b) - F(a)

**w/o interval : ... + c

Basic Trigonometric & Logarithmic Integrals

∫ sin(x)dx = -cos(x) + c ∫ cos(x)dx = -sin(x) + c

∫ tan(x)dx = -ln |cos(x)| + c

∫ sec(x)dx = ln |tan(x)+sec(x)| + c

∫ ln(x)dx = xln(x) - x+ c

Advanced Trigonometric Integrals

∫ dx / sqrt(1 - x^2) = arcsin(x) + c

∫ dx / 1+x^2 = arctan(x) + c

∫ dx / |x| sqrt(x^2 - 1) = sec^-1 (x) + c

*******or arcsec(x) + c

Area Under The Curve PT1 (AUCPT1)

a-b∫ f(x) dx = area under f from a to b

*AREAS ARE ALWAYS POSITIVE

**if f(x) is negative on [a,b], Area under curve f(x) between a,b = -(a->b)∫ f(x) dx

Area Under the Curve PT2 (AUCPT2)

Suppose f(x) >= g(x) on [a,b],

-Area between f(x), g(x) on [a,b] is (a-b)∫ f(x) dx= (a-b)∫ (f(x) dx -g(x))dx

(a-b)∫ f(x) dx = (a-c)∫ f(x) dx + (c-b)∫ f(x) dx

*if a < c < b

Average Value of f(x) on the interval [a,b] =?

1/(b-a) * integral from a to b of f(x) dx

Growth

If k > 0, we have exponential growth

y' = ky

y= yi * e^kt

-yi=initial amount

-k=rate of increase

-t=number of time periods

Decay

If k < 0, we have exponential decay

y'= ky

y= yi * e^kt

-yi=initial amount

-k=rate of increase

-t=number of time periods

Unit Circle

Linear Approximation

L(x) = f(a) + f'(a)(x-a)

Volume of Sphere And Its Derivative

V = 4/3πr³ : V'=4πr^2 * dr/dv

Area of Circle

A=πr²

Derivative of An Inverse

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

Equation of Line Tangent To a Graph

1. find y-value when x is given

2. f'(x)

3. find slope at f'(y-value of given x)

4. point slope form - y-y1 = m(x - x1)

Population Growth Formula

P=Ne^rt or

P(t)= P(initial) * e^rt

-> P(t) is the population at time t

-> P(initial) is the initial population size

-> r is the growth rate

-> time