Final Flashcards
A limit only exists when
The lim from the right = lim form the left
A limit DNE when
Function has a different height from the left and right side of x = c, vertical asymptotes, or if f(c) oscilates between 2 fixed values as x approaces c
Process for solving limits
1) Direct Substitution
2) 0/k means lim = 0
3) n/0 means lim DNE
4) 0/0 means do more
Methods for 0/0 limits
Factor, Foil, Rationalize (cojugate), getting rid of complex fractions, etc. Always go back to direct sub to finish the problem
lim as x approaches 0 of sin(x)/x or sin(kx)/kx
1
lim as x approaches 0 of 1 - cosx / x
0
lim as x approaches 0 of tanx/x
1
Definition of continuity (3 rules)
1) f(c) is defined
2) lim x->c f(x) exists
3) lim x->c f(x) = f(c)
Intermediate Value Theorem
If f(x) is continuous on the closed interval [a,b] and k is some height between f(a) and f(b), then there exists some c value between [a,b] such that f(c)=k
Slope of the Secant Line
AROC (y2-y1)/(x2-x1)
Slope of the Tangent Line (Derivative)
IROC
f(x) is not Differentiable When
discontinuous, corner, cusp, or a vertical tangent, different slopes from the left and right
f(x) is Differentiable When
1) Continuous
2) Slope from the right = slope from the left
Power Rule f(x) = x^n
f’(x) = nx^(n-1)
Product Rule d/dx [f(x)g(x)]
f(x)g’(x) + f’(x)g(x)
Quotient Rule d/dx [f(x)/g(x)]
[g(x)f’(x)-f(x)g’(x)]/g(x)g(x)
lo di hi - hi di lo over lo lo
Chain Rule (Most Important)
d/dx [f(g(x))]
f’(g(x)) g’(x)
Work from the outside and work your way in only one derivative at a time
d/dx sinx
cosx
d/dx cosx
-sinx
d/dx tanx
sec(x)^2
d/dx cotx
-csc(x)^2
d/dx secx
sec(x)tan(x)
d/dx csc
-csc(x)cot(x)
Implicit Differentiation
Derive both sides with respect to x. When taking the derivative of a term with y attach a y’ (chain rule). Gather all the y’ on one side and solve for y’
Derivatives of Inverse Functions at a point (p,q)
The original function is (q, p)
If g(x) is the inverse of f(x)
Then g’(p) = 1/f’(q)
Equation of the Tangent Line
y = m(x-x1) + y1
where m is the slope of the tangent line (derivative) and (x1, y1) is the point of tangency
Tangent Line Approximation
Find the tangent line equation of the given x. Plug in estimate x value into tangent line equation
Related Rates
1) Draw a picture
2) Write down the Find and Know
3) Write an equation that matches the variables
4) Differentiate both sides of the equation with respect to time
5) Plug in given information and solve for the needed rate. DO NOT PLUG IN ANYTHING UNTIL AFTER YOU HAVE TAKEN THE DERIVATIVE!
6) Circle the final answer and give the UNITS
Volume of a Sphere
4/3 pi r^3
Volume of a Cone
1/3 pi r^2 h
Volume of a Cylinder
pi r^2 h
Critical Numbers
f’(x) = 0 or undefined
First Derivative Test
Finding Increasing and Decreasing Intervals
Test the intervals between the critical numbers to determine if f’(x) is positive or negative
f’(x) > 0 means f(x) is
Increasing
f’(x) < 0 means f(x) is
Decreasing
Relative Max Occurs When
The derivative of the function changes from positive to negative at a critical number
Relative Min Occurs When
The derivative of the function changes from negative to positive at a critical number
Absolute Max and Mins (closed interval)
Find f(x) values for endpoints and critical numbers; determine the highest and lowest values
Second Derivative Test
Uses concavity to determine if a critical number is a relative max or min
f’(c) = 0 and f’‘(c) > 0 (Positive)
f has a relative min at x = c
f’(c) = 0 and f’‘(c) < 0 (Negative)
f has a relative max at x = c
When f’‘(x)>0
f(x) is concave up
When f’‘(x)<0
f(x) is concave down
Point of Inflection
A point where the graph of a function changes concavity
f’‘(x) has to change signs in order for there to be an inflection point
Mean Value Theorem
If the function f(x) is continuous on [a,b], AND differentiable on (a,b), then there is at least one number x=c in (a,b) such that
f’(c) = f(b) - f(a) / b - a
The tangent slope equals the secant slope
Intermediate Value Theorem
If a function f(x) is continuous on [a,b] and N is a value between f(a) and f(b), then there exists a value c in (a,b) such that f(c) = N
Extreme Value Theorem
If f(x) is continuous on [a,b] then f(x) has both an absolute max and min on [a,b]
Fundamental Theorem of Calculus Part 1
If f(x) is continuous on [a,b] and F(x) is an antiderivative of f(x) then
integral from a to b of f(x)dx = F(b)-F(a)
Top - Bottom
Fundamental Theorem of Calculus Part 2
If F(x) = the integral from a to x of f(t)dt then F’(x) = f(x)
Rolle’s Theorem
If f(x) is continuous on [a,b], differentiable on (a,b), and f(a) = f(b) then there exists a c in (a,b) such that f’(c) = 0
L’Hopital’s Rule
If the lim x->c of f(x)/g(x) results in an indeterminate form and f(x) and g(x) are differentiable then
lim x->c of f(x)/g(x) = lim x->c of f’(x)/g’(x)
Area of a Square (Cross Section)
b^2
Area of a rectangle with h = 1/2b (Cross Section)
1/2 b^2
Area of an Isosceles Right Triangle with base as leg (Cross Section)
1/2 b^2
Area of an Isosceles Right Triangle with hypotenuse as base (
Cross Section)
1/4 b^2
Area of an Equilateral Triangle
(Cross Section)
sqrt(3)/4 b^2
Area of a Semicircle (Cross Section)
pi/8 b^2
