math

Derivative of rootx+ rooty=c

-root y/x

range of sec(x) on the interval (0, pi/2)

1 to infinity

quadratic formula

A bad boy (-b) couldn't decide whether or not (+-) to go to a radical party. But he was being a square (b2), and get turned down by 4 awesome chicks. And it was all over by 2 AM. For ax2+ bx +c = 0

Vertex

A quadratic function can be graphed using a table of values. The graph creates a parabola. The parabola contains specific points, the vertex, and up to two zeros or x-intercepts. So, with a, b, c, changes, one can a series of graphs with changing shapes.

Which graph has a range that include all real numbers? sinx, cosx, tanx, secx

Answer tanx remember the curve

y=3x2+4x-5, what is the slope of the tangent line at point (3, 4)?

Answer: 22. Slope is the derivative, y'=6x+4, plug in x=3, y'=22=m

f(x)=[x+1/x-1]3, f'(2)=

Answer: 54 chain (let an function =x+1/x-1), then 3u2u' (general power rule) and quotient (for u') f'=3(x+1/x-1)2(x+1/x-1)' after arrangement and plug in x=2, f'=54

Inverse trigonometric functions

Arc- are used to obtain an angle from any of the angle's trigonometric ratios. Can use -1 or arc, in the unit circle, "the arc whose cosine is x" is the same as "the angle whose cosine is x". Also, only part of curve is shown here, remember they are continuous, but on y axis rather than x axis. arcsec because curve open to x axis

Y-intercept of y=tan-1(x)

Arctanx, tanx rotate for 90 degree, still the 0,0 point. Series of lines across the Y axis instead of X axis as tanx.

Radians

Degree is actually abritually set to 360 a circle, degrees, technically speaking, are not actually numbers, and we can only do math with numbers. That is why we have to convert it to radians, 2pi=360 pi=180 Radian=2*angle

A perfect snowbell with a diameter of 4 feet is melting at a rate of 8pi ft3/hr. At what rate is the radius decreasing?

Differentiation with respect to time gives a rate of change. Implicit differentiation is used to solve related rates problem. Answer: 0.5 General derivative, that is derivative with y rather than x. V=4/3pi r3, the rate of decrease is derivative to time, so dV/dT (4)=8 is given, dV/dT=pi4/3x3r2r'=4r2r' plug in r=2, 16r'=8 r'=0.5 (over T). Actually surface of ball is derivative of V, 4pir2

Where is y=x/x2+1 increasing?

Function is increasing where f'>0, while function is concave up, f' is increasing, that is f''>0. Find f' use quotient rule, so, (-1,1)

Inverse function

Geometrically, a function and inverse function have graphs that are reflections, in the line y = x. This reflection operation turns the gradient of any line into its reciprocal. Their two derivatives, assuming they exist, are reciprocal. note 1/x3 roughly same shape as 1/x.

Plane equation that passing though (-1,2,3) and perpendicular to vector i+j-2k.

Here the point and normal vector is known, just plug to i(x-x1)+j(y-y1)+k(z-z1)=0 where x1,y1,z1 is the point and ijk is the normal vector. Or can be interpreted as dot product with normal vector should be 0, that is the definition of normal vector. (i=1)(x+1)+(j=1)(y-2)+(k=-2)(z-3)=0 X+Y-2Z=-5"

Where is y=x3+3x concave down?

If f '(x) is increasing, then the function is concave up and if f '(x) is decreasing then the function is concave down. To determine whether the derivative is increasing or function concave, we take the second derivative. A function is concave down when second function is < 0. y'=3x2+2, y''=6x, 6x<0, so range (-infinity, 0)

Find tangent line equation to x2y-y3=8 at (3, 1)

Implicit differentiation and product rule, x2y=2xy+x2y' 2xy+x2y'-3y2y'=0, plug in point value x=3, y=1, yield y'=-1 line-point formula, x-3=-y+1 or x+y=4

Area bound by y=2x, y=x, x=1, x=4.

Integer, also one can use addition and subtraction with function [2x-[x=[2x-x=[x, so just definite integration of y=x between 1 and 4 x2=15/2

IMPLICIT DIFFERENTIATION

It is important to note that the derivative expression for explicit differentiation involves x only, while the derivative expression for implicit differentiation may involve BOTH x AND y .What you are derived to, usu to x, so, others like y should always use general rule

Line equation

Lines written the slope-intercept form: y = mx + b, m is the slope, b is the y-intercept (x=0). Point-Slope form Equations y - y0 = m(x - x0), m is the slope, x0, y0 is the value of the point that we know. Vector-Parametric form which defines series of lines in 3D, there are 2 points and a t, x=(x1, y1)+t(x2, y2) , ration of y/x in the parametric term is the slope. slope=y2/x2.

Mean Median Mode

Mean is average, median the middle number or average of two middle number, mode is the most frequent number

Find SD of 1, 2, 5, 8, 9., If all value increase by 2, the new mean and SD.

Mean=25/5=5 n=5. Every value-mean square: 16 (-4²), 9, 0, 16 , 9 equals to 50 Square root of 10. Mean+2=7, SD the same.

Standard deviation

Note: within 1 sd, actually means between +1 and -1 sd, sometime need to /2 or find the average to calculate value. SD is square root of addition of (every value-mean)2/population number. More sample, large sample size, less deviation SD, the closer average to expect number, more accurate (approve of theory). Beware of approve or disapprove

Plane equation or Cartesian equation of a plane

One need a normal vector (i, j, k, the normal is perpendicular to any line in the plane) and a point (x1, y1, z1). Plane equation would be i(x-x1)+j(y-y1)+k(z-z1)=0, rearrange to general form ax+by+cz=d

Probability

Probability is the ration of desired vs other outcome. If attempts is ordered, "or" denotes "added" probability in one attempt, "and" denotes multiply probability in multiple attempts; if no order, then need to multiply attempt number. Permutation probability is n!. or use n!PdQo/D!O! equation. All probabilities adds up to 1.

radical graph

Radical cannot take negative value.

fx=x2+2x+1, sum of 4 1/2 width rectangle areas under the curve (upper left touch the curve). 6.77, 8.66, 10.75 or 17?

Rectangle approximate which is always less than real integration. Integ to x3/3+x2+x+c plug in 2 and 0, 8/3+6=8.66, but must be smaller, so, 6.77.</div>"

Derivative rule: product rule

Remember the rule in the following way. Each time, differentiate a different function in the product and add the two terms together.

Discontinuity

Removable: a discontinuity that can be "repaired" by filling in a single point. In other words, a removable discontinuity is a point at which a graph is not connected but can be made connected by filling in a single point. Discontinuity: jumpy. Asymptote, or a graph that gets closer and closer to a line but never hits it. The reason is because there are typically x values (domains) where the function or graph does not exist at all, since we cannot divide by 0. One of the simplest rational functions, the inverse function, is y = 1/x, as x gets larger and larger, y gets closer and closer to 0. For y = 1/x, we have a horizontal asymptote at y = 0 and a vertical asymptote at x = 0. For piecewise function, check boundary points, equal, then continuous.

Rise run

Rise, change in Y, while run is change in X. "Sine is first, rise is first" meaning that Sine takes the angle of the line segment and tells its vertical rise. "Cosine is second, run is second" meaning that Cosine takes the angle of the line segment and tells its horizontal run. "Tangent combines the rise and run". This shows the main use of tangent and arctangent: converting between the two ways of telling the slant of a line, i.e., angles and slopes.

Derivative rule: Quotient rule

THE QUOTIENT RULE Remember the rule in the following way. Always start with the "bottom'' function and end with the "bottom'' function squared. Note that the numerator of the quotient rule is identical to the ordinary product rule except that subtraction replaces addition.

y=ekx-2 where K>0. As K increases, what happens to x or y intercept?

There are similar questions, what the vertex will change etc. One can use real number to test. For this question easily see y intercept will never change y=e0-2=-1. X intercept, actually ekx-2=0, ekx=2 so kx is a constant In2 and inversely correlated, K increase x should decrease.

f(x)=x3, tangent line slope is 5, find the x value on the line

This means f'(x) is already know, f'=3x2=5 x=+-(5/3)1/2

Tangent line equation

To find an equation for a line, all we need is the slope at that point, and the coordinates of a single point on that tangent line. 1. Take the derivative of your function. 2. Plug in your x-value, x0, into the derivative to get your slope. 3. Use x0 to find out what the y-coordinate of that point is. Basically, just plug in x0 to get f(x0) = y0. Then that point, (x0, y0), is a point on your tangent line. 4. Use point-slope formula to write the equation out: y - y0 = m(x - x0).

Derivative rule: Chain rule

Usually, we invoke an intuitive approach. For example, it is sometimes easier to think of the functions f and g as "layers'' of a problem. Function f is the "outer layer'' and function g is the "inner layer.'' Thus, the chain rule tells us to first differentiate the outer layer, leaving the inner layer unchanged (the term f'( g(x) ) ) , then differentiate the inner layer (the term g'(x) ).

y=x2-Kx+1, K>=o, as K increases, the vertex will change this way

Vertex is the max/min point or y'=0 y'=2x-K=0 x=k/2 (K increase, x must also increase) y=1-k2/2. The point (k/2, 1-k2/2) is vertex, clearly, K increase, x increase while y decrease.

vertical asymptote of y=x/(x2-4)

X cannot be +2, or -2, that is VA which means curve can approach but never reach these 2 points.

h(t)=30(sinpit/6), h changing rate when t=2

derivative is the changing rate or instant speed or tangent line etc. chain rule Dt=30(cospit/6)(pit'6)=5picospi/3=pi5/2 cos60 0=1/2 Whenever chain rule, general rule.

Evaluate [3,2,4] [2-1,0]

dot product 4. Just a single value

Derivative rule: power rule

general power rule Ina, if ex derivative still ex since Ine=1 by definition

1/2log25+log3/5+log4=

log12

Inverse trigonometric derivative: arcsecx &amp; arccsc

same thing arccsc just add - sign"

Sphere volume

surface 4pir2, sphere is countless pyramid add up together, hence the 4/3.

y=3e2x, the slope of tangent line is

y'=6e2x=2y, slope of tangent line directly proportional to y.

1. s(t)=2t2+3t+4, what is the instantaneous velocity at t=2; 2. s=cubic root t, t=8 what is instantaneous velocity.

1. 11, 2. 1/12, s, position, velocity is s', acceleration is s''

Graph rationals

1. Factor to see if any removable discontinuities (or holes) exist; cross out on top and bottom. To get the y value of the hole, you can use the crossed out version and plug in the x value. You know that part of the curve of the graph goes close to that point, but you have to graph a small circle there. 2. Draw any VA asymptotes from setting anything left in the denominator to 0. (You may get none, but there can be more than one.) 3. Draw any EBA (HA or oblique) asymptotes (You may get none, but there will be at most one). 4. Determine the x intercepts (where y = 0), and y intercepts (where x = 0). 5. (More Advanced) See if the function crosses any horizontal asymptotes by setting the original function equal to the HA. Solve for x; you already have the y (from the asymptote). 6. Draw "T charts" to fill in extra "key" points, for example, on the sides of the EBA asymptotes. 7. Domain is everything except where the removable discontinuities or asymptotes exist. 8. Have your graphs "hug every asymptote", but remember that you will never have more than one point on a vertical line, since we're drawing functions.

Absolute value of complex number

absolute value of a complex number is the number's distance from the origin in the complex plane (2 sides of right triangle).

f'(x)=3x²+2x and (1,3) lies on the curve f(x), what is f(2)?

answer 13. Integration or anti-derivative f(x)=x3+x2+C plug in value, C=1, now plug in x=2, y=13.

gx=(fx)3, f(0)=-3, f'(0)=2, find g'(0)

answer 54 use chain rule and general power rule g'=3u2u' where u=f(x), g'=3(fx)2f', for g'0, plug in f(0)=-3, f'(0)=2.

Find a vector perpendicular to [2,-1,3]. a.[-2,1,-3] b.[1/2,-1.1/3] c.[3,-1,2] d. [0,3,1]

answer d. To perpendicular, dot product is 0, use this to check. Just a single value. X1x2+y1y2+z1z2, if equal to 0, perpendicular.

derivative cos2(5x)

answer: -10cos(5x)sin(5x). Here need to use chain rule twice, one have to use general derivative every time (sinu=cosuu')

derivative x<2+y2=1

answer: -x/y Implicit

f(x)=arcsin(2x2), find f'(1/2)

answer: 4/root3 chain rule u=2x2 also always use general rule

Integration rectangle method

approximate method, divide into rectangles and add their area together Or definite integral between two values. If the leftmost line of rec is on the cure, the approximate must

Inverse trigonometric dx: arcsinx and arccosx

arccosx just add - sign, since the addition of derivative of arcsin and arccos is 0.

Domain of function sinx/x+1, Domain of root 4-x2

x<-1 or x>-1 removable discontinuous, as long as x+1 not 0. This curve is like oscillating spring shape. Here 4-x2 must at least 0 so, x2=4. Domain or discontinuity, two major issue denominator cannot be 0; minus number cannot be rooted