Calculus 2 Midterm 2 Cheat Sheet (DRAFT) by bryxe

First, Look for potential u-subs­tit­ution and potent­ially remove the du from original function. If the degree is an odd number, look at removing one term to manipulate the remaining term because it is a multiple of 2. Lowering the degree from 2 to 1 is possible using trig identi­ties. Be observant of rewriting terms ex (1/___)

There are three types of Improper Integrals. Type 1: Integrate from 1 to Infini­ty/­-In­finity. Type 2: Integrate from 0 to 1, typically improper at zero, or possibly identi­fying where denomi­nator is zero. Type 3: is a combin­ation of both. Type 1 converges on the P-Series (1 / x^p) when P > 1. Diverges when P<= 1. Type 2 converges when P < 1, diverges when P >= 1. Now when solving the integral look for u-subs­tit­ution and changing the x-bounds to u-bounds. The u may be typically found in the denomi­nator, but look for the du to replace. In the case that there is a ln, you will need to do Integr­ation by Parts, then complete. When completing Type 1 look to take the limit as t->inf, and replacing inf with t in the bounds. Be aware of Trig Identities as well. Type 2 look for vertical asympt­otes. Similarly, there will be u-sub needed. In the bounds split at the V.A and solve as it approaches the value from the left, and from the right. Remember changing the x-bounds to u-bounds if substi­tuting.

When given x = f(t) and y = g(t) use the formula, where alpha is less than t and beta is greater than t. First find the derivative of x, and the derivative of y, this is done how we have done in the past. Once we have both first deriva­tives, square them. This will allow us to plug into the formula and plug in our bounds. After doing so, look for u-subst­itu­tion. If not easily access­ible, take out the GCF inside the radical. Now we can split into two radical times each other. For example, rad(144t²+144t⁴) = rad(144t²(1+t²) = rad(144t²) times rad(1+t²). Then look for u-subst­itu­tion. Be aware of the changing bounds, from t to u. The compute anti-d­eri­vative. Also, look for (A+B)² formula to simpify.

Look at Formula. If x = f(t) and y = g(t). Once finding the derivative of f(t), substitute into the formula, where bounds are in terms of t. Then, you may be able to distri­but­e/s­implify by multip­lying (FOIL). Then compute the anti-d­eri­vative. Use similar techniques mentioned to compute anti-d­eri­vative.

First identify which identity to use. [Trig Identi­ties]. Then find the value of a^2 and a. Plug a into the formula selected, and then calculate the deriva­tive. Solve the derivate for dx, and substitute x and dx back into the original function. Manipulate the radical to potent­ially rewrite using Trig Identi­ties. If there is no obvious substi­tution, you will most likely need to complete the square. Once you have completed the square look to use the formula. Then follow the same steps as above. Be careful of the bounds in u-subs­tit­ution.

Identify which Formula to Use. Referring to x-values but switch to y-values if about the y axis. Identify the bounds of the integrals, x-values we are integr­ating from. Then use f(x) and find the first deriva­tive. After finding the first deriva­tive, square it. This will allow you to substitute inside the radical in the formula. A simpler integral will allow u-sub inside the radical, and then compute the anti-d­eri­vative. Be aware of whether or not you are changing the bounds. In other situations manipulate the radical, remove the [f^I(x)]² term, and multiply it by its reciprocal to get 1, and then just 1 so it is equal to itself. For example, (1+x^-2/3) = x^-2/3(x^2/3+1). Then due to properties of radicals, you can remove the part multip­lying and so you have the radical(x^-2/3) times radica­l(1+x^2/3). Then simplify and complete using u-subst­itu­tion. Be aware of changing bounds and Trig Identi­ties.

Table Method, setup t, x(t), and y(t). Then solve for different values of t and sketch on a cardinal plane. Obtain Regular x/y equation Harder, and you solve for t in terms of the x equation and then substitute that into the y(t) formula. So you have y in terms of x not t.

Look for the Formula. Setup using point slope form, y-y1 = m(x-x1) and needing a Point(­x1,y1). First calculate the deriva­tives of x(t) and y(t). Then have the y(t) derivative as the num. and derivative of x(t) as the denom. This is the slope or m. If you are given a t value, you may solve for an x1 point by substi­tuting t into x(t). Likewise for y(t). Now you have a point on the curve and can plug into point-­slope form. You can also substitute t into your slope formula, (dy/dx), to obtain the numerical slope, finishing your tangent equation. You can distribute and manipulate into y=mx+b form. If you are given a point, set the x1 value = x(t), and solve for all solutions of t. Do the same with y1 value given = y(t). The t value that is shared is the t used in the slope equation (dy/dx). Then obtain m and plug into point-­slope form. Horizontal The numerator of the slop (m) = 0. Obtain the t value that is a zero, then using that t value substitute into x(t) and y(t) to get the P(x1,y1) the tangent occurs at. Vertical Same thing except find zeros of the denomi­nator. Finding Slope, set the m equal to the desired slope. Then find t, plug into x(t) and y(t) for the point.

First check the degree of the numerator and denomi­nator. If the degree of the denomi­nator is greater than the numerator you may begin Partial Fraction Decomp­osi­tion. If Otherwise complete long division and begin P.R.D on the remainder. Factor the denomi­nator so all solutions are present. Then setup P.F.D by doing separate integrals on the solutions and adding the integrals together. If the solution is liner (x+1), (x+2), (x+2)², then the numerator is just a constant letter, A. If the solution is non-liner, (x²+1), then the numerator is Cx+D, where C and D are constants. Then complete the anti-d­eri­vative, and pay attention to log rules.

Look at X-axis Formula. Alternate 1. First identify what y = _. then R(x) = y, so R(x) is the function. Now that we have the function f(x), compute the first deriva­tive. After doing so, square the first deriva­tive. The answer will be substi­tuted into the formula we are using. After substi­tuting into the formula, look for u-subst­itu­tion. Then compute the anti-d­eri­vative. If you have two sqrt function multip­lying each other, then you can multiply the inside. Simplify after multip­lying, and either preform the anti-d­eri­vative or look for u-sub. Pay attention to changing bounds if using u-sub. If they gives x in terms of y LOOK HERE. Here you will use the same function a little different. The radius is the distance between the curve and the axis-o­f-r­ota­tion. So here we use Alternate 2 and R(y) = y. Then solving is the same from above just in terms of y. The bounds here are y-values.

Look at Y-Axis Formulas. Alternate 1. If given y in terms of x and the range in x-values, we want the radius in terms of x. So R = x. F(x) is the function given. Compute the first derivative of F(x), then square the result. This will allow us to substitute into our formula. Initially, look for u-sub inside the radical, if it is possible to cancel du on the outside, do so and compute the anti-d­eri­vative. Pay attention to changing the bounds when completing u-sub. Alternate 2. If given the range in terms of y, solve x in terms of y. R = x = F(y). The function is now in terms of y. Now compute the first derivative of F(y), and square the result. This can now be substi­tuted into the formula. Initially, look for u substi­tution to replace inside the radical and cancel the function it is multiplied by. Pay attention to changing the bounds if doing u-sub.