I don't quite understand how to go about solving these so if you could explain them It would be much appreciated! ^o^
1. The radius, r, of a sphere is increasing at a constant rate of 0.05 meters per second.
A. At the time when the radius of the sphere is 12 meters, what is the rate of increase in its volume?
B. At the time when the volume of the sphere is 36π cubic meters, what is the rate of increase in its surface area?
C. Express the rate at which the volume of the sphere changes with respect to the surface area of the sphere (as a function of r).
2. A particle moves along the x-axis so that its velocity at any time t > 0 is given by
v(t) = (2π − 5)t −sin(π t).
A. Find the acceleration at any time t.
B. Find the minimum acceleration of the particle over the interval [0, 3].
C. Find the maximum velocity of the particle over the interval [0, 2].
3. Consider the function h(x) = a(−2x + 1)^5 − b , where a ≠ 0 and b ≠ 0 are constants.
A. Find h′(x) and h′′(x).
B. Show that h is monotonic (that is, that either h always increases or h always decreases).
C. Show that the x-coordinate(s) of the location(s) of the relative extrema are independent of a and b.
4. Consider g(x) = {a sin x+b, if x < or equal to 2π}
and {x^2 - πx +2, if x > 2π}
A. Find the values of a and b such that g(x) is a differentiable function.
B. Write the equation of the tangent line to g(x) at x = 2π.
C. Use the tangent line equation from Part B to write an approximation for the value of g(6). Do not simplify.
The key to all these is the chain rule.
v = 4/3 πr^3
dv/dt = 4πr^2 dr/dt
so, when r=12m,
dv/dt = 4π * 144 * 1/20 = 144π/5 m^3/s
the surface area
a = 4πr^2, so when v=36π, 4/3 πr^3 = 36π. r = 3
da/dt = 8πr dr/dt = 8π * 3 * 1/20 = 6π/5 m^2/s
you want dv/da. That is just
(dv/dt) / (da/dt) = (4πr^2 dr/dt) / (8πr dr/dt) = r/2
You can go through a lot of algebra coming up with v(a), but this is a lot easier.
#2
v(t) = (2π − 5)t −sin(π t)
the acceleration is just a(t) = dv/dt = (2π-5) - πcos(πt)
minimum acceleration is where da/dt = 0. That is
π^2 sin(πt) = 0
maximum v is where dv/dt = 0
But since you are working with an interval, make sure that the local max (where dv/dt=0) is bigger than v(t) at the ends of the interval.
#3
h(x) = a(−2x + 1)^5 − b
h' = 5a(-2x+1)^4 * -2 = -10a(-2x+1)^4
h" = -10a * 4(-2x+1)^3 * -2 = 80a(-2x+1)^3
(B) since h; is always positive or negative, depending on the sign of a.
(-2x+1)^4 is always positive
(not quite. at x = 1/2, h'=0 so at that instant h is neither increasing nor decreasing)
(C)the roots of h' do not depend on a or b.
#4
is that a sinx + b
or a sin(x+b)?
In any case, make sure that g(x) and g'(x) match on both sides of x=2π
Then, since you have both g(x) and the slope at x=1π, use the point-slope form of the tangent line.
plot both g(x) and the tangent line on some handy plot site, such as desmos.com to verify your answers.
Then use the line with x=6 to approximate g(6), since 6 is close to 2π
The format is a bit confusing but I think it is a sinx + b :)
also, I'm really sorry Reiny. I have a weak immune system and deal with a lot of hereditary health issues so I've missed quite a bit of my calc lessons. Right now we are mainly learning all the fundamentals or "building blocks" so I'm trying to use these practice problems to see how everything is solved so I can reference back to them while I study. I know it was a lot to ask for and I'm sorry for not being specific with what I was having problems on and instead just coming off like a dumb kid trying to cheat on their test.
If I were to list my main problem it would be the application of the rules. I know the basic rules but I have trouble connecting which ones to use with the problems. If you have any advice that could help with this I would appreciate it very much. I'll also try to work through the problems again and come back with the places I get stuck! Thank you both so much for your help so far
the best way to learn the application of rules is to work a lot of exercises. You will learn which rules apply in which situations.
google is a great resource. If you search for a problem, you will usually find discussions and solutions.
Certainly! I'll go through each question and explain how to approach solving them.
1. The radius, r, of a sphere is increasing at a constant rate of 0.05 meters per second.
A. To find the rate of increase in the volume of the sphere, we need to differentiate the volume formula with respect to time. The volume of a sphere is given by V = (4/3)πr^3, where r is the radius. Taking the derivative of this equation with respect to time (t) gives: dV/dt = (4/3)π(3r^2)(dr/dt). Given that dr/dt = 0.05, and we need to find the rate of increase when r = 12, substitute these values into the equation to find the rate of increase in volume.
B. To find the rate of increase in the surface area of the sphere, we need to differentiate the surface area formula with respect to time. The surface area of a sphere is given by A = 4πr^2. Taking the derivative of this equation with respect to time (t) gives: dA/dt = 8πr(dr/dt). Given that dr/dt = 0.05, and we need to find the rate of increase when A = 36π, substitute these values into the equation to find the rate of increase in surface area.
C. To express the rate at which the volume of the sphere changes with respect to the surface area, we divide the rate of change of the volume (found in part A) by the rate of change of the surface area (found in part B).
2. A particle moves along the x-axis so that its velocity at any time t > 0 is given by v(t) = (2π - 5)t - sin(πt).
A. To find the acceleration at any time t, we need to differentiate the velocity function with respect to time (t). The velocity function v(t) is given as v(t) = (2π - 5)t - sin(πt). Taking the derivative of this function v'(t) will give you the acceleration at any time t.
B. To find the minimum acceleration of the particle over the interval [0, 3], evaluate the acceleration function from part A at the critical points and endpoints of the interval. Find the value of t where the acceleration is minimized.
C. To find the maximum velocity of the particle over the interval [0, 2], we need to find the maximum value of the velocity function v(t) over that interval. Evaluate the velocity function at the critical points and endpoints of the interval and find the maximum value of v(t).
3. Consider the function h(x) = a(-2x + 1)^5 - b, where a ≠ 0 and b ≠ 0 are constants.
A. To find h'(x), the derivative of h(x), apply the power rule and chain rule when differentiating the function. To find h''(x), differentiate h'(x) once again.
B. To show that h is monotonic (always increasing or always decreasing), examine the sign of the first derivative h'(x). If the first derivative is always positive or always negative, then the function is monotonic.
C. To show that the x-coordinate(s) of the location(s) of the relative extrema are independent of a and b, examine the critical points of h(x) by finding where the first derivative h'(x) equals zero or is undefined. Show that these critical points do not depend on the values of a and b.
4. Consider g(x) = {a sin x + b, if x ≤ 2π} and {x^2 - πx + 2, if x > 2π}.
A. To ensure that g(x) is differentiable, we need to check for continuity at x = 2π. This means that both parts of the function (sin x + b and x^2 - πx + 2) should have the same value at x = 2π. Equate the values of both parts of g(x) at x = 2π and solve the resulting system of equations to find the values of a and b.
B. To write the equation of the tangent line to g(x) at x = 2π, we need to find the derivative of g(x) and evaluate it at x = 2π. Use the derivative to find the slope of the tangent line, and then use the point-slope form of a line to write the equation of the tangent line.
C. To approximate the value of g(6) using the tangent line equation from part B, plug x = 6 into the equation of the tangent line. Do not simplify the expression to obtain the approximation for g(6).
Remember, in many of these problems, you will need to use calculus techniques such as differentiation and the rules of derivatives to find the answers. These explanations should provide you with a starting point for each question, but be sure to check your work and solve each question step by step.
#1 you have to recognize certain words and phrases in these type of questions.
"radius, r, of a sphere is increasing at a constant rate of 0.05 meters per second"
the key words are "rate" and m/s
and I conclude that I could rephrase that as : dr/dt = .05 m/s
you want to find "the rate of increase in its volume", that to me means: find dV/dt
so to have a dV/dt, I must first have a V = ...
But we know for a sphere, V = (4/3)π r^3
and we want dV/dt and have a dr/dt, we should find the derivative with respect to t
dV/dt = 4π r^2 dr/dt
we want this when r = 12, so all you have left to do is some arithmetic
b) This time you want dV/dt when V = 36π
so (4/3)π r^3 = 36π
r^3 = 27
clearly r = 3
now you are at the same stage as in a)
Btw, nobody is going to do this whole test or assignment for you
I did the first one, but no more.
Show me what steps you have so far for the others, what your thoughts are on
proceeding, and what your major problems are.
Most of these questions are of a fundamental nature and you MUST know how
to do them