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Browse High School Calculus
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Chain rule.
    Maximizing the volume of a box.
    Maximizing the volume of a cylinder.
    Volume of a tank.
    What is a derivative?

Derivatives [12/07/1998]
Can you write x^2 = x+x+...+x (x times) if x is not a positive integer?

Derivatives and Antiderivatives [10/31/2001]
When I run the formula int(sqrt(cos(x^2)),x=0..1) in Maple, it just returns the formula. Why?

Derivatives, and the Big Deal with Little Intervals [04/07/2015]
A teen struggles with the concept of instantaneous rates of change. Throughout several rounds of clarification and analogies, Doctor Ian underscores how remarkably well calculus models natural phenomena.

Deriving Arctan without Using Numbers [03/22/2002]
How do you do you derive arctan without using numbers, or how do you prove (arctan)' = 1/1+x^2?

Deriving Simpson's Rule [06/07/1998]
Can you show me a derivation of Simpson's Rule?

Deriving the Area of a Sphere [10/21/2003]
I know the area of a sphere is 4phi(r^2), but I'm wondering how to derive that formula. I know it should be done in cylindrical coordinates, and I'm thinking that the arc of a circle is defined as rd(theta) and it's multiplied with rd(phi) to get (r^2)d(theta)d(phi). Could you please help explain this?

Deriving the Gamma Function [12/15/2000]
How can you prove that sqrt(pi)/2 = (1/2)!, and what is a fractional factorial like that equal to?

Deriving the Integral for the Surface Area of a Sphere [06/28/2001]
I need to find the cost per square foot of steel that makes up a tank, so I need a way to derive the surface area of part of a sphere with a given diameter...

Deriving the Quotient Rule [9/5/1995]
How do I prove this derivation: (f/g)' = fg'-fg'/g^2 ?

Deriving Trig Functions; Taylor Series [05/01/2001]
How would I find, from first principles - no tables, no calculator - for example, 32 degrees? If I use a formula, how is it derived?

Descartes' Method for Tangents [09/02/1998]
Can you help me find the equation of the tangent of y^2 = 2x using Descartes' method...?

Design a More Efficient Soda Can [4/23/1995]
The problem is to design a more efficient soda can that holds the regular 12 oz. of liquid. The can needs to have the least possible surface area.

Determining Tangent to an Ellipse, Minimizing Area [02/20/1998]
Find the equation of the tangent to the ellipse that forms, with the coordinate axes, the triangle of smallest possible area.

Determining the Length of a Coil of Ribbon [08/31/2001]
How will calculus give me the same answer I found with algebra and geometry?

Determining Whether a Function is Continuous [06/10/2000]
How can you tell whether or not a function is continuous?

Differences in Differentiation Notation? [06/06/2013]
A calculus textbook variously represents derivatives as dy/dx, d/dy, d/dx(f(x)).... What does it mean to differentiate "with respect to x" anyway? Doctor Peterson makes sense of the many notations, starting with the basics.

Differentiability, Intervals, Inflection Points of Piecewise Function [05/13/1998]
For what values of k and p will the function be continuous and differentiable? On what interval will it be increasing? Find all points of inflection.

Differential Equations and Flow Rate [02/14/1999]
A solution containing 3 lb/gal of dye flows into a 100 gal tank at 5 gal/min ...

Differential Notation [04/16/2009]
Why is the second derivative written as d^2y/dx^2?

Differentials [09/18/1998]
What is the difference between finding the derivatives of a function (dy/dx), and finding its differentials (dy,dx)?

Differentiate Twice [11/14/1997]
If y = xsin(3x) prove that y''+9y = 6cos(3x), where y'' is its second derivative.

Differentiating [01/14/1998]
Please help me differentiate (y^2) * cos (1/y) = 2x + 2y.

Differentiating and Integrating the Formula for Area of Circle [05/11/1998]
The formula for a circle's circumference is the derivative of the formula for its area. What is the significance of this?

Differentiating a Polynomial [04/25/1999]
Differentiate with respect to x: f(x) = x^8 + 3x^5 + (3-5x)^4.

Differentiating the Ceiling Function [10/16/2000]
How do I find dD/dp for an equation involving the ceiling function? How do I minimize D as a function of P?

Differentiating the Rate of Decay [3/14/1996]
A lump of radioactive subtance is disintegrating at time 't' days after it was first observed to have mass 10 grams and: (dm/dt) = -km (where k is a positive constant). Find the time, in days, for the substance to reduce to 1 gram in mass, given that its half life is 8 days.

Differentiating Under the Integral Sign [01/11/2001]
Can you give me an example of the integration method called "differentiating under the integral sign"?

Differentiating y with Respect to ... y? [11/05/2010]
A student familiar with differentiation struggles to take the derivative of a function with respect to the self-same variable. Doctor Ali puts the student back on track with a clearer substitution.

Differentiating y = x^x [11/3/1994]
Please could you differentiate y=X^X (that's X to the power of X)?

Differentiation Problem [11/15/1997]
A light shines from the top of a pole 50 ft. high. A ball is dropped from the same height at a point 30 ft. away from the light...

Distance From a Point to a Plane [03/31/1998]
Can you show me the proof of the formula for the distance between a point and a plane?

Distance to the Sun [04/16/1999]
Find the distance from Earth to the sun when t = 90 days...

Does f Have a Local Extrema at x = 0? [07/10/2003]
f(x) = (x^3)/6 + (x^2)/2 + cosx - 1. Does f have a local extremum at x = 0?

Domain, Asymptotes, Intercepts of a Function [04/01/2003]
What is the domain of this function? What asymptotes does it have? What are the x and y intercepts? Etc...

Domain/Range of a Function [01/22/1997]
How do you find the domain and range of the function f(x) = 2x^2-3x+1? (Both with and without calculus.)

Dominant Terms [03/25/1998]
What are dominant terms, and how do you obtain their values?

Donkey Grazing Half a Field [08/08/1997]
A donkey is attached by a rope to a point on the perimeter of a circular field. How long should the rope be so that the donkey can graze exactly half the field?

Double Integration in Polar Coordinates [03/27/2003]
Evaluate double integral x-y/x*2+y*2 over x*2+y*2 equal to or less than 1.

e as a Series and a Limit [03/30/1998]
Why does e = 1 + 1/2! + 1/3! + 1/4! + ... and lim (1 + 1/n) ^ n, as n -- > infinity?

An Easy Definition of Calculus [9/4/1995]
What is calculus?

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