It has been something like 5 years since I've done any calc and so I was wondering if there were any smart people who can answer these questions >_>
Let f(x) be the function defined by f(x) = 2x^3 - 3x^2 - 12x + 20
(a) Write an equation of the line normal to the graph of f(x) at x=0
(b) find the x- and y- coordinates of all poitns on the graph of f(x) where the line tangent to the graph is parallel to the x-axis
(c) sketch the graph of f with the line(s) tagnent to the point(s) that you found from part b.
I'm trying to wrestle through this and like 8 other problems and I really haven't touched the subject in too long. Does anyone know how to do this stuff still?
a) the derivative of f at 0 is -12, so the normal line has equation y = x/12 + c, and since f(0) = 20, c=20.
b) a line is parallel to the x-axis when it has 0 derivative, so you want to find the points at which the derivative of f is 0. The derivative of f is a quadratic, so just use the quadratic formula.
c) f is a cubic, and from b) you know the stationary points, so sketch it from this. Add on the lines you found in b).
OOPS I can't read, it said normal not tangent. a) The line will be y = -(1/f '(0))*x+f(0), since it has to go through f(0) and have slope perpendicular to f '(0). b) Set the derivative equal to 0, solve for x, plug into f(x) to get the corresponding y values
added spaces because those primes are sorta hard to see
-- No I'm not a damn furry. Looney Tunes are different. - Guiga I wanted Sonic/Shadow romance at that time, not sex. - MWE
Also because the coefficient of the x^3 term is positive, your graph should be going towards negative infinity on the left and towards positive infinity on the right. Basically, it should like a stylized 'N'.
-- Worry is a misuse of imagination. Nicely done, SuperNiceDog. Super nicely done.
i'm taking physics this fall and so I'm going to need to spend some real time this summer re-learning how to do calculus. Does anyone have a good suggestion on how to do that? Maybe there is a site explaining calc I could work through? I'm sure as long as I'm starting from scratch I'll pick it up much easier the 2nd time as things come back to me. I remember terms like derivative and whatnot, but I can't recall how the different notations cause the number and functions to interact.
But basically I have learned that I will be absolutely no help to my younger brother on his calculus final. (It's a take home final)
Does anyone want to solve and show work on like 7 problems for him? <__<
If not that's cool, he should have paid more attention in class if he can't do it either after a whole semester.
Will bookmark that for later. I'll probably start learning calc again in July or something. I did finish 2 semesters of calculus (got an A and a B) and so I know that I was really good at it once. But apparently when you don't do something for over 5 years you can forget how! <__<
Sure, I don't mind answering some questions if they're as brief as the ones in the OP.
Whelp I dunno how much work is involved in these, but if you wanted to answer these problems then at the very least he could use them to compare his own work too. (aka he's probably just going to want to copy it >_<'' )
2) The position of a particle is given the equation s(t) = (1/3)t^3 - 8t^2 + 28t where t is measured in seconds and s in meters.
Use Calculus to describe the details of the particle's motion for 0 <= t <= 5
A) Find the velocity and acceleration functions.
B) Graphs all three functions: Position, velocity, and acceleration.
C) When is the particle speeding up? When is it slowing down? Give a reason for your answers.
3)
Let g(x) = [integral symbol with x at the top and -1 at the bottom] f(t) dt
A) graph the function f(x) = (1/4)x + 2 FOR -2 <= x < 4 8 FOR x = 4 2x-5 FOR 4 < x <=6 -x+13 FOR 6 < x <= 9
B) Computer G (3) and G' (-12)
C) Determine the instantaneous rate of change of g, with respect to x, at x=2
D) What is the absolute minimum value of g on the interval [-2,4]. Justify your answer.
Another helpful website is http://math.stackexchange.com/questions It has a huge database of questions, so there's a good chance of finding something in there that's very similar to your problem.
-- ~Peaf~ "Later that day, I f***ed a panda." - PSO
Let v(t) be the velocity in feet per second of a skydiver at the time t seconds, t >= 0. After her chute opens, her velocity satisfies the differential equation dv/dt = -2(v+17), with the initial condition v(0) = -47
a) Use separation of variables to find an expression for v in terms of t where t is measured in seconds.
b) Terminal velocity is defined as lim v(t) (as t approaches infinity). Find the terminal velocity of the skydiver to the nearest foot per second.
C) It is safe to land when here speed is 20 feet per second. At what time t does she reach this speed?
(5)
Let f(x) be the functiond efined by f(x) = cos^2(x) - cos(x) for 0 <= x <= (3%u03C0/2)
a) determine the exact values of the x-intercepts of the graph of f(x) b) use calculus to determine the intervals on which f(x) is increasing. c) find the x-values for which the tangent to f(x) is parallel to the horizontal axis.
a) v = ds/dt, a = dv/dt, i assume he knows how to differentiate polynomials.
b) probably best to sketch them in reverse order. a(t)=2t-16 is just a line so that's easy. v(t) is a quadratic with obvious roots (2 and 14) and one turning point (where a is 0, ie t=8), so sketch it from this. then, as in the question in the OP, s is a cubic with positive leading coefficient and you know its turning points so sketch it based on these.
c) it is speeding up when a>0, so t > 8. slowing down when a < 0, so t < 8.
7) Continuous means the limit must be the same at every point from both directions. In these cases you just need to pick k such that the two piecewise parts are equal at the points where the two parts meet (so x = 3, x = 4 for the two problems).
8) V = (4/3)pi R^3 dV/dt = 4 pi R^2 dR/dt (by chain rule) dV/dt = 100 ft^3 / min
A) let R = 5, solve for dR/dt B) SA = 4 pi R^2 dSA/dt = 8 pi R * dR/dt
Plug in, solve for dSA/dt.
--
No amount of planning will ever replace dumb luck.
a) f is linear on each segment so this is easy to graph.
b) g(3) is just the integral of a linear function, hopefully he will know how to do this. my answer is 9. bit confused about how to find g'(-12) since iirc g' = f but f is only defined between -2 and 9 so we don't know what f(-12) is, maybe someone else has some input.
c) this is g'(2) = f(2)
d) g' = f is positive on [-2,4] so g is increasing on this interval, and thus the minimum is g(-2).
a) dv/(v+17) = -2 dt so integrating on both sides yields ln(v+17) = -2t + c. exponentiate both sides to get v in terms of t, then use the initial condition to get a particular solution. i get v = 30exp(-2t) - 17.
b) exp(-2t) tends to 0 as t tends to infinity, so the answer is -17.
b) differentiate f (use chain or product rule for the first term). the intervals on which f is increasing are exactly those intervals on which f'(x) => 0, so find these.
c) tangent is parallel to horizontal axis when it has 0 derivative, so this reduces to solving f'(x) = 0.
From: Justin_Crossing | #023 man they take roundabout ways of saying 'find the local minima and maxima' It's Calc 1. That's all they learn how to do! Haha.
Asking the questions in a roundabout way does make sure they understand the ideas behind it though.
--
No amount of planning will ever replace dumb luck.
If you're just taking Physics 1/2 I'm pretty sure if you have a decent understanding of working derivatives and doing relatively simple integration you should be fine math wise.
That said I'm taking it this summer so this is just second hand info >_>
-- Claims Umineko no Naku Koro Ni's goat furniture.
I didn't read the responses, but I see there are enough of them that surely somebody has gotten it right. I'm mostly replying to let you know that I make my living as a math tutor, so feel free to ask me questions in the future.