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15 Cards in this Set
- Front
- Back
- 3rd side (hint)
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What is derivative of 1 / (3x^3 + 4x +8)? |
-(9x^2 + 4) / (3x^3 + 4x +8)^2. It is essentially x^(n) with n being -1. so derivative is n*x^(n-1) dx. |
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When is quadratic equation and completing the square useful in calculus? |
I can use either of those to find the intercepts with this x-axis. Completing square is when you set y=0. Bring C over to right side of =. Then add the number to both sides that completes the square function. (If it helps can always divide both sides by number in front of x^2 term to help with completing square) |
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What are critical points of functions? What is first derivative and second derivative use in regards to these? |
Inflection pts, minima and Maxima. Critical pts are when the first derivative is equal to 0. Second derivative value helps determine if it is a min, max, or inflection pt. |
First derivative =0 solved for x finds the critical points. The second derivative let's us know if those points are minima, Maxima, or inflection pts. |
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What is test for a maximum? Minimum? Inflection pt? |
Max: 1st der = 0. 2nd der < 0. Min: 1st det = 0. 2nd der > 0. Inflection pt: 2nd der = 0. |
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What is sharpness? What are eqs for sharpness? What is radius of curvature? How do I find eqs? |
Sharpness is represented by K and is an instantaneous representation of a point on a given curve represented by a function. K = y'' / [1+(y')^2]^(3/2) x'=dx/dy K = -x'' / [1+(x')^2]^(3/2) R represents the radius of curvature. It is the absolute value of reciprocal of K. R = 1 / |K| Eqs found by searching curvature. |
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What is Lhopitals rule? |
It is that the limit of F(x)/g(x) = limit of F'(x)/g'(x). This is true for the 2nd, 3rd, etc derivatives of the functions. Useful to get rid of dividing by 0. |
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How do you find centroid of area with area is given as a function? |
Use given eq. A = integral(F(x)dx) xc = integral (xdA) / A yc = integral (ydA) / A dA = F(x)dx = g(y)dy note: bounds for A and xc will be on x axis. yc bounds should be on axis. Can be hard to find if area is not bounded by two x = or y= functions. |
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What is symbol for first moment of an area? What is it a function of? |
M. My = integral(xdA) = Xc*A Mx = integral(ydA) = Yc*A |
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What is moment of inertia? What is parallel axis theorem? What is d? What is polar moment of inertia? |
Is the second moment of an area. Ix = integral (y^2dA) Iy = Integral (x^2dA). dA=f(x)dx. Moment of inertia of parallel axis = moment of inertia about centroidal axis + Ad^2. d is distance between the two axises. J = integral(r^2dA) = Ix + Iy |
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What is gradient vector? What is divergence of a given vector field? So what does divergence of a vector field V represent? How do i find eq? |
Gradient vector gives the maximum rate of change if the scalar function. Eq: gradient vector = derivative (I term respect x)i + derivative( J term with respect to y)j + derivative( k term with respect to z)k.
They accumulation of substance in a small region or a substance flowing at a point. Divergence is equal to dot product of gradient vector and vector
Basically if that dot product = 0 then the substance isn't accumulating and it is incompressible.
Basically search divergence. |
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What is the curl of a vector field? What if curl is 0? What is formula? |
It is the cross product between gradient vector and that vector. Meaning it terms how fast the flux is rotating and in what direction. Flow is irrotational. Look up curl. It is cross product between V and gradient vector. |
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What is the Laplacian function? How do i find eq? When is it used? |
Second derivatives of given function f(x,y,z) with respect to x,y,z added together. Lapalacian of a scalar function is also know as divergence of the gradient function. Search gradient. Used in voltage potential, gravitational potential, concentration gradients, pressure gradients, and thermal gradients. |
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Where can I find list of derivatives in NCEES? What are derivatives of cos and sin? |
Search derivatives or calculus. Early in calculus part of mathematics section. d(sinu)/dx = cosu*du/dx d(cosu)/dx = -sinu*du/dx |
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(This is diff eq question) What is a difference eq? How do I find eq? |
It is a discrete version of the original differential eq. This is used when exact solution may be difficult to obtain. It is a relationship between a function and it's differences over some interval of integers.
Search difference equations. |
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What formulas can be used to find ao, an, and bn in Fourier series theorem? When are these helpful? |
an = (2\T)*integral[f(t)*cos(n*wo*t)*dt] bounds 0 to t ao = (1/T)*integral(f(t)dt) bounds 0 to t bn = (2/T)*integral[f(t)*sin(n*wo*t)*dt] bounds 0 to t
When asked to find the first few terms of Fourier series. |
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