Vector Calculus Mid-Term #1 WebAssign Study Guide

Let f(x,y) = xe^(x²-y) and P=(#,#). (a) Calculate ||∨fp||. ||∨fp|| = __________________ (b) Find the rate of change of f in the direction of ∨fp. _____________________ (c) Find the rate of change of f in the direction of a vector making an angle of 45° with ∨fp. ________________________

(a) - Find fx and fy and set in vector format - Plug in P values for x and y - Find vector magnitude (b) - Same answer as (a) (c) - |∨fp|*|u|*cos(θ) - the magnitude of u is always going to be 1 -|∨fp|*cos(θ)

Compute the curvature k(t) of the clothoid r(t)=<x(t),y(t)> where x(t)= integral from 0 to t of sin(u^#/#)du and y(t)= integral from 0 to t of cos(u^#/#)du. k(t) = ___________________

- Answer is t^(#-1)

If limit from t→t₀ of r(t)=u, then which of the following is true? MC

The limit from t→½t₀ of r(2t)=u

Find an arc length parameterization of the line y=mx+b r1(s) = __________________

- x=t - r(t)=<t,mt+b> = <0,b> + t<1,m> - r(s) = convert "t" to s/√(1²+m²)

Calculate the gradient. f(x,y) = cos(x^#+y) ∨f = ______________________

- ∨f = <fx, fy>

Calculate the gradient. h(x,y,z) = xyz^# ∨f = ___________________

- ∨f = <hx, hy, hz>

Compute the length of the curve over the given interval r(t)=< , , > and # <= t <= # L = ________________________

-Find r'(t) -Find |r'(t)| - L is the integral of |r'(t)| using the given interval

Find the critical points of the function given f(x,y). (x,y) = ( ___ ) max/min/saddle (x,y) = ( ___ ) max/min/saddle (x,y) = ( ___ ) max/min/saddle

- Find fx, fy, fxx, fyy, fxy - Use fx or fy to solve for x or y to create the number of critical points that are expected in the question - D(x,y) = fxx*fyy - (fxy)² - Plug in critical points to D - If D < 0, it is a saddle point - If D = 0,no conclusion - If D > 0 and fxx > 0, it is a local minimum - If D > 0 and fxx < 0, it is a local maximum

Let r(t) = < , , >. Calculate r'(t), T(t) and evaluate T(#). r'(t)=___________________ T(t)=___________________ T(#)=__________________

- Find r'(t) - T(t) = r'(t)/|r'(t)| - T(#) = plug in # to T(t)

Find T, N, and the tangential and normal components of a at the specified value of t r(t) = < , , > and t = # T = ___ N = ___ aT = ___ aN = ___

- Find r'(t) and then r'(#) - Find r"(t) and then r"(#) - Find |r'(#)| - Find r'(#) (dot) r"(#) - Find r'(#) x r"(#) - Find T = r'(t)/|r'(t)| - Find N = T'(t)/|T'(t)| - Find aT =r'(#)(dot)r"(#)/|r'(t)| - Find aN = r'(#) x r''(#) / |r'(t)|

Find arc length parametrization of r(t)=< , , >. r1(s) = _________________________

- Find r'(t) and then |r'(t)| - Take the integral of |r'(t)| from 0 to t to get s=#t and then get t = ln |(s/#)+1| -Plug in new t to r(t)

Express the arc length of y=10x^2 for 0<=x<=R as an integral using the parametrization r(t)=<t^#,#t^#> but do not evaluate it s(t) = ∫ from 0→4√(3) of ______________

- Find r'(t) and then |r'(t)| and then plug in the |r'(t)|

Calculate the curvature function for r(t) = < , , > k(t) = ___________________

- Find r'(t) x r''(t) - Find |r'(t) x r''(t)| - Find |r'(t)| -k(t) = |r'(t) x r''(t)| / |r'(t)|³ -If t is given, plug in at end

Find the point on the graph of z = #x^2-4y^2 at which vector n = <n1, n2, n3> is normal to the tangent plane. (x, y, z) = ( _________________ )

- Find the gradient, which is equal to <x', y', z'> and is noraml to the surface - So, we now need our gradient to be equal to the normal vector given - <x', y', z'> = k<nx, ny, nz> for some k - x'=n1*k, y'=n2*k, z'=n3*k -Solve the z'=n3*k for k - Plug in k to find x and y -Plug in x and y values into original equation to find z

Find a parameterization of the tangent line at the indicated point r(t)=(ln(t))i+(t^-#)j+(#t)k, t=#1 L(t) = _____________________

- L(t) = <r(#1)> + t<r'(#1)> - Answer in form of < , , >

Let r(t) = < , , > Find T, N, and B at the point corresponding to t=#. T = ____ N = ____ B = ____ Find the equation of the osculating plane at the point corresponding to t = 1 __________________________________

- T = r'(t)/|r'(t)| - N = T'(t)/|T'(t)| - B = T(t) x N(t) - Osculating: B1(x-x0)+B2(y-y0)+B3(z-z0)=0

Find the normal vector N(t) to r(t)=<#, sin(#t), cos(#t)> N(t) = _______________________

- T(t) = r'(t)/|r'(t)| - N(t) = T'(t)/|T'(t)| Answer: <0,-sin(#t),-cos(#t)>

Find an equation of the tangent plane to the graph of f(x, y) at the point (#,#)

- Tangent Plane Equation n1(x-a)+n2(y-b)+n3(z-c)=0 - Find fx(x,y) and fy(xy) - Find fx(#,#) and fy(#,#) - z=fx(#,#)(x-#)+fy(#,#)(y-#)

Determine the global extreme values of the function on the given set without using calculus. f(x,y) = ####, #<=x<=# and #<=y<=# maximum ________________ minimum ________________

- The boundaries of x and y give four possible extreme boundaries - Plug these points into f(x,y) and the min is the lowest result while the max is the highest result

Estimate f(x#dec,y#dec) given that f(x,y) = #, fx(x,y) = x#, and fy(x,y) = y#. f(x#dec,y#dec) = ________________

- The equation to use is : f(x,y) + fx(x,y)*(d#x - x) + fy(x,y)*(d#y - y)

Determine the global extreme values of the function for the given domain f(x,y) given: x, y >= 0, x+y <= 1

- The global maximum is at (1,0) for all f(x,y) expressions - The global minimum is 0 for all f(x,y) expressions

Compute the partial derivatives

- The variable in the denomintor is the variable you take derivatives with respect to - Every other variable is treated as a constant

Determine the global extreme values of the function on the given domain given f(x,y) and y >= x - #, y >= -x - #, y <= # maximum _______________ minimum _______________

- Usue the given y-value and plug it into the other comparisons to solve for the x-values - Use the x values and plug them into the comparisons they *didn't* come from to get the remaining y-value - Make new expression of #<=x<=# and #<=y<=# and plug them into f(x,y) to get maximim and minimum

Find the acceleration vector a(t) for a particle moving around a circle of radius #r cm at a constant speed of v = #v cm/s. a(t) = _______________________

- a(t)=-rw²<cos(wt),sin(wt)> - w = v/r - -rw²= -v²/r - Plug r and w into a(t)

Find aT & aN as a function of t given r(t) = < , , > aT = ______________________ aN = _____________________

- aT = r'(t) (dot) r''(t) / |r'(t)| - aN = r'(t) x r''(t) / |r'(t)|

Find an arc length parameteriation of the circle in the plane z=#z with radius #r and center (#1,#2,#z). r(g(s)) = ____________________

- r(t)=<#rcost+#1,#rsint+#2,#z> - Find r'(t) and |r'(t)| - Take integral of |r'(t)| from 0 to t to get s=#t and then get t=s/# -Plug in new t into r(t)

Calculate the directional derivative in the direction of v at the given point. Remember to normalize the direction vector. f(x,y) = ######, v = <#,#>, P = (#,#) Duf(P) = _____________________

- u = v/|v| - ∨f(x,y) = <fx,fy>*u - Plug in given P points for x and y to get Duf(P)

Calculate the velocity and acceleration vectors and the speed at the time indicated. v(0) = ___ a(0) = ___ vs(0) = ___

- v(t) = r'(t) - a(t) = r''(t) - vs(t) = |v(t)| which is just v(t) without the ijk

Find the velocity vector v(t) given the acceleration vector a(t) and the initial velocity. a(t) = < , , >, v(0) = < , , > v(t) = ____________________

- v(t) = ∫a(t) - Remember the +C - Compare v(t) with v(0) to solve for C

Find r(t) and v(t) if given a(t), v(0), and r(0) r(t) = ___________________ v(t) = ___________________

- v(t) = ∫a(t) - Remember the +C - Compare v(t) with v(0) to solve for C - r(t) = ∫v(t) - Compare r(t) wth r(0) to solve for C

What is the domain D of r(t)=(e^t)i+(t^#)j+((t+#)^#))k? (Enter your answer using interval notation.) D = _____________________

-Find the domain of each individual component -Combine them to make one final domain -If it is an infinity or borders a union, use a "(" -If it isn't, use a "["

Find the speed at the given value of t r(t)=< , , > and t = #

-Find the |r'(t)| and then plug in t to get answer

Evaluate the limit... lim(t→0) of r(t)/t for... r=<sin(#t),1-cos(#t),#t> _____________________

-First, divide each component by t -Then take the limit of each new component -Enter answer as < , , >

Evaluate r(#1) and r(#2) for r(t)=<sin(πt/2),t²,(t²+9)⁻¹> r(#1)= _________________________ r(#2)= _________________________

-Plug in #1 for t and enter -Plug in #2 for t and enter

The helix with radius R and height h and N complete turns has the parameterization r(t)=<Rcos(2πNt/h),Rsin(2πNt/h),t>, 0≤t≤h Calculate the lenght of a helix if R=#, h=# and N=# _______________________________

-Plug in all given numbers -Find r'(t) -Find |r'(t)| -Take the integral from 0 to h

Evaluate the derivative where r1(t)=<t^#,t^#,t^#> and r2(t)=<e^#t,e^#t,e^#t> d/dt(r1(t)*r2(t)) = _____________

-Product rule -If "t" value is given, plug into result

Evaluate the derivative where r1(t)=<t^#,t^#,t> and r2(t)=<0,e^t,0) d/dt(r1(t) x r2(t)) = ____________

-Take the cross product -Then take derivative

Compute the tangent vector for the cycloid r(t)=<t-sin(t),1-cos(t)> at t=# r(#) = _________________

-Take the derivative of both components -Plug in t value -Enter in vector notation

Compute the derivative for r(t)=(cos(#t))i+sin^#(t))j+(tan(#t))k dr/dt=__________________

-Take the derivative of each component and enter it in either vector form or with the ijk in front

Find ∫<(1/u^#),(1/u^#),(1/u^#)>du from 1/2 to 1

-Take the integral for each individual component -Enter answer in vector format < , , >

The function r(t)=(#cos(t))i+(#sin(t))j traces a circle. Determine the radius and center of the circle. r = ____________ center = (_________) Determine the plane containing the circle (MC)

-r = # -center = 0,0,0 unless there is an obvious translation -The plane is xyz minus the component (xyz or ijk) that has a zero in it

Which of the following is an arc length parameterization of a circle of raduus 7 centered at the origin? MC

Answer: r(t) = <#sin(t/#), #cos(t/#)>

Find the critical point of the function f(x,y) = √(####). Determine whether it is a minimum, maximum, or saddle point. (x,y) = ( ____ ) max/min/saddle

Answer: (0,0) minimum

Which of the following functions will NOT describe the same curve as the one that is displayed? MC

Answer: -r(t)

Which of these curves is farther from the origin than the one that is displayed? MC

Answer: 2(rt)

Each tangent plane touches this surface at P and at how many other points?

Answer: Infinitely many points

Determine the global extreme values of the function on the given set without using calculus. f(x,y) = e^(-x²-y²), x²+y²<=#

Answer: Max = 1 Min = e^-#

Suppose that the limit from t→t₀ of ||r(t)|| = ||u||. Then what can be said about the assertion limit from t→t₀ r(t) = u? MC

Answer: The assertion may be correct

Suppose that C is a curve contained in the surface that passes through point P... MC

Answer: The only option that has "contained" in it Remember "Question with 'contained' has the answer 'contained'"

What is the magnitude of the vector T+2N+3B? MC

Answer: √(#)