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Selected answers to common questions:
Maximizing the volume of a box.
Maximizing the volume of a cylinder.
Volume of a tank.
What is a derivative?
- Trigonometric Functions and Euler's Identity [04/10/1998]
We need to prove three trigonometric functions in various ways.
- Trigonometry and the Chain Rule [12/01/1998]
I need help differentiating expressions such as: y = 2 csc^3(sqrt(x)) and
y = x/2 - (sin (2x))/4. Where does the chain rule fit in?
- A Trigonometry Integral Requiring Two Substitutions [05/03/1998]
Substituting twice and using trig to integrate sqrt(1 + sin(x)).
- Triple Integrals [12/07/1997]
Determine the limits...
- The Troublesome Endpoints of a Trigonometric Series [04/10/2011]
A student has doubts about the radius of convergence assumed in a proof that
involves integrating a power series. Doctor Vogler confirms his suspicion about the
behavior of endpoints of integrated series, and introduces Abel's Theorem.
- Trying to Integrate f(x) = exp(-ax^2) [11/23/2001]
How would you integrate a function of the form f(x) = exp(-ax^2)?
- Tuning a Piano [05/28/2002]
Each piano has a unique tuning curve that can be approximated through
3-point quadratic interpolation. Is there a formula that will allow me
to measure three different notes on the piano and find the tuning
curve in between these notes?
- Two Points Moving... [7/18/1996]
Point P is moving in the positive direction along the y axis at a
constant velocity of 10 m/s. Point Q is always moving with a constant
speed of 11 m/s directly towards point P...
- Two Somewhat Similar Problems [1/9/1995]
I need help with two similar limit of integral problems.
- Two Ways to Find a Formula [04/14/1998]
I need to show that Sigma(rx^r) = (x-(n+1)x^(n+1)+nx^(n+2))/(1-x)^2.
- Two Ways to Integrate [04/15/1998]
How do you integrate: Int[sin^4(x)*cos^6(x)]dx?
- Undefined and Indeterminable ... at the Same Time? [09/05/2010]
A student wonders whether the labels "undefined" and "indeterminate form" could
apply to one and the same expression. Doctor Vogler considers several expressions,
functions, and limits to distinguish the different contexts that call for such terminology.
- Understanding the Need for Limits [02/19/2001]
Can you explain limits in a way that makes the need for them clearer? Do
limits apply to all functions, or are they only useful for certain ones?
- Unit and Basis Vectors in Three Dimensions [05/09/1998]
Explanations and uses of unit vectors and basis vectors.
- Using Differentiation to Find Derivatives [11/16/2002]
Our class would greatly appreciate it if you could simplify the whole
idea of using differentiation to find derivatives, and give a real-world
- Using L'Hopital's Rule [01/22/1999]
Prove that e^(-1/3) is the limit for y =(arctan(x)/x)^(1/(x^2)) when x ->
- Using Newton's Method to Solve an Implicit Equation [10/14/2007]
How can I solve the equation (sin(X))/X = 0.7031? My book gives the
answer but does not explain how to find it.
- Using the Chain Rule: Define Functions [5/2/1995]
What are the integrals of sin(ln(4x+5)) and ln(sinx) ?
- Using the Chain Rule: Finding a Derivative [2/26/1996]
d/dx sin(sin (2cos(2x/3))) = ?
- Using the Definition of Derivative [9/1/1995]
Using the definition of derivative, find f'(x), in simplest form, for
f(x)=x^(1/3). (I know through the "tricks" that the answer is (1/3)*x^(-
2/3) but I can't get to that using the definition of derivative.) If
f(x)= [x], prove or disprove that f(x) has a limit at x=1.
- Using the Power Rule to Find the Derivative [9/24/1995]
We are trying to prove that the "power rule" works for finding the
derivative of x^(3/2) (where x>0).
- Vector Analysis [7/6/1996]
(1) Given three points A,B,C, find the angle between R(AB) and R(AC); the
(scalar) area of triangle ABC; a unit vector perpendicular to ABC...
- Vector Calculus [2/8/1996]
Find the total distance travelled by a particle along the path, sketch
the path; find the equation of the tangent line at t1, sketch the curve
and tangent at t1...
- Vector calculus [11/18/1994]
I have been studying vector calculus, and I was interested in finding
more about the principles of the divergence theorem and Stokes' theorem.
- Vector Calculus [02/10/1999]
Prove the following identity: a . (grad(a . v) - curl (v x a)) = div v...
- Vector Equations Of Lines [6/28/1996]
A boat leaves a harbour, O, position vector (0i+0j) at 9 a.m... at what
time will it arrive?
- Visualizing 1 + 1/x [10/10/2003]
Show that the sum of a positive number and its reciprocal is at least 2.
- Volume of a Cone or Pyramid [03/30/1998]
Proofs that the volume of a cone or pyramid is (1/3)b*h.
- Volume of an Elliptical Cone [10/15/1998]
Can you help me on find the volume for an elliptical cone by using a
- Volume of a Right Circular Cone [10/07/2001]
Using calculus, derive the formula for the volume of a right circular
cone with a radius of r and height h.
- Volume of a Rotated Region [11/02/1998]
Find the volume of a solid formed by revolving the region bounded by
y=x^2+1, y=0, x=0, and x=1, about the y-axis.
- Volume of a Shape [11/11/1997]
What is the volume of the shape formed by rotating the parabola y=x^2
around the line y=x? (From x = 0 to 1).
- Volume of a Solid [10/02/1997]
The base of a solid is the region inside the circle x^2 + y^2 = 4...
- Volume of a Sphere [04/21/1998]
Can you help me derive and prove the formula for the volume of a sphere?
- Volume of a Torus [04/22/1999]
I would like to know how to find the volume of a torus using integrals.
- Volume of Intersecting Pipes [10/27/2000]
How can you calculate the volume of the intersection of two perpendicular
pipes of the same radius?
- Volume of Spherical Cap [02/06/1998]
I am trying to find the volume of a cap of a sphere with radius of 5. The
cap has a height of 3 - it is as if the top of the sphere, 3 meters from
the top, was severed from the rest of the sphere.
- Volume Using Cross Sections [05/28/2001]
Find the volume of the region between y = |x| and y = -|x|+6 with cross
sections that are equilateral triangles and perpendicular to the x-axis.
- Washers and Discs, Graphs and Symmetry [2/20/1996]
Use the washer method to find the volumes of the solid generated by
revolving the regions bounded by the lines and curves about the y-axis...
- Water Draining from a Conical Reservoir [12/20/1995]
A circular conical reservoir, vertex down, has depth 20 ft and radius of
the top 10 ft. Water is leaking out so that the surface is falling at the
rate of 1/2 ft/hr...