Calculus

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I recommend this book: A Course of Modern Analysis by Whittaker and Watson. You may also find this book at Google Books. This book is a hundred years old and is considered the classic calculus book.


Derivatives

Definition of Derivative

, provided the limit exists.

For the following problems, find the derivative using the definition of the derivative.

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Power Rule



For the following problems, compute the derivative of with respect to

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solution Derive the power rule for positive integer powers from the definition of the derivative (Hint: Use the Binomial Expansion)

Product Rule



For the following problems, compute the derivative of with respect to

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solution Derive the Product Rule using the definition of the derivative

Quotient Rule



For the following problems, compute the derivative of with respect to x

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solution Derive the Quotient Rule formula. (Hint: Use the Product Rule).

Generalized Power Rule



For the following problems, compute the derivative of with respect to x

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Chain Rule



For the following problems, compute the derivative of with respect to x

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Implicit Differentiation

For the following problems, compute the derivative of with respect to x

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solution where is a function of


Logarithmic Differentiation

For the following problems, compute the derivative of with respect to x

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solution for any functions and where

Second Fundamental Theorem of Calculus



For the following problems, compute the derivative of with respect to x

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solution Give a proof of the theorem

Applications of Derivatives

Slope of the Tangent Line

solution Find the slope of the tangent line to the graph when .

solution Find the slope of the tangent line to the graph when .

solution Find the slope of the tangent line to the graph when .

solution Find the equation of the tangent line to the graph when .

solution Find the slope of the tangent line to the graph when .

solution Find the slope of the tangent line to the graph at the point .

solution Find the equation of the tangent line to the graph at the point .


Extrema (Maxima and Minima)

solution Find the absolute minimum and maximum on of the function .

solution Find the absolute minimum and maximum on of the function .

solution Find all local minima and maxima of the function .

solution Find all local minima and maxima of the function .

solution If a farmer wants to put up a fence along a river, so that only 3 sides need to be fenced in, what is the largest area he can fence with 100 feet of fence?

solution If a fence is to be made with three pens, the three connected side-by-side, find the dimensions which give the largest total area if 200 feet of fence are to be used.

solution Find the local minima and maxima of the function .

Related Rates

solution A clock face has a 12 inch diameter, a 5.5-inch second hand, a 5 inch minute hand and a 3 inch hour hand. When it is exactly 3:30, calculate the rate at which the distance between the tip of any one of these hands and the 9 o'clock position is changing.

solution A spherical container of meters is being filled with a liquid at a rate of . At what rate is the height of the liquid in the container changing with respect to time?


Projectile Motion

Integrals

Riemann Sums

Integration by Substitution

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Integration by Parts



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Trigonometric Integrals

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Trigonometric Substitution

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Partial Fractions

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Special Functions

solution A ball is thrown up into the air from the ground. How high will it go?

solution Let be a continuous function for .
Show that

solution Evaluate

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Applications of Integration

Area Under the Curve

solution Find the area under the curve on the interval .

solution Find the total area between the curve and the x-axis on the interval .

solution Find the area under the curve on the interval .

solution Using calculus, find the formula for the area of a rectangle.

solution Derive the formula for the area of a circle with arbitrary radius r.

Volume

Disc Method

The disc method is a special case of the method of cross-sectional areas to find volumes, using a circle as the cross-section.

To find the volume of a solid of revolution, using the disc method, use one of the two formulas below. is the radius of the cross-sectional circle at any point.

If you have a horizontal axis of revolution



If you have a vertical axis of revolution



solution Find the volume of the solid generated by revolving the line around the x-axis, where .

solution Find the volume of the solid generated by revolving the region bounded by and around the x-axis.

solution Find the volume of the solid generated by revolving the region bounded by and around the x-axis.

solution Find the volume of the solid generated by revolving the region bounded by and around the y-axis.

solution Find the volume of the solid generated by revolving the region bounded by , the x-axis and the line around the x-axis.

Shell Method

Cross-sectional Areas

solution Find the volume, on the interval , of a 3-D object whose cross-section at any given point is a square with side length .

solution Find the volume, on the interval , of a 3-D object whose cross-section at any given point is an equilateral triangle with side length .

solution Find the volume of a cylinder with radius 3 and height 10.

Arc Length




solution Calculate the arc length of the curve from to .

solution Determine the arc length of the curve given by x = t cos t , y = t sin t from t=0

solution Calculate the arc length of y=cosh x from x=0 to x

Mean Value Theorem

solution Find the average value of the function on the interval .

solution Find the average value of the function on the interval .

solution Find the average speed of a car, starting at time 0, if it drives for 5 hours and its speed at time t (in hours) is given by .

solution Find the average value of the function on the interval .

solution Deduce the Mean Value Theorem from Rolle's Theorem.

Series of Real Numbers

Sequences

nth Term Test

If the series converges, then .

Note: This leads to a test for divergence for those series whose terms do not go to but it does not tell us if any series converges.

solution Discuss the convergence or divergence of the series with terms .

solution Discuss the convergence or divergence of the series and .

solution Discuss the convergence or divergence of the series .

Telescopic Series

If is a convergent real sequence, then .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

Geometric Series

The series converges if and, moreover, it converges to . For any other value of r, the series diverges. More generally, the finite series, for any value of .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

Integral Test

If the function is positive, continuous, and decreasing for , then

and

converge together or diverge together.

Notice that if were negative, continuous, and increasing this is also true since such a function would simply be the negative of some function which is positive, continuous, and decreasing and multiplying by -1 will not change the convergence of a series.

solution Discuss the convergence of .

solution Discuss the convergence of .

solution Discuss the convergence of .

solution Discuss the convergence of .

solution Explain why the integral test is or is not applicable to .

solution Explain why the integral test is or is not applicable to .

solution Explain why the integral test is or is not applicable to .

solution Discuss the convergence of .

solution Discuss the convergence of .

solution Discuss the convergence of .

Comparison of Series

Direct Comparison
If for all
If converges, then also converges.
2. If diverges, then also diverges.
Limit Comparison
Suppose , and where is finite and positive.
Then and either both converge or both diverge.

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

Dirichlet's Test

Let for .

If the sequence of partial sums is bounded and as , then converges.

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series where .

solution Discuss the convergence or divergence of the series for any integer .

solution Discuss the convergence or divergence of the series where .

Alternating Series

If , then the alternating series

and

converge if the absolute value of the terms decreases and goes to .

solution Discuss the convergence of the series .

solution Discuss the convergence of the series .

solution Discuss the convergence of the series .

solution Discuss the convergence of the series .

solution Discuss the convergence of the series .

solution Discuss the convergence of including whether the sum converges absolutely or conditionally.

solution Discuss the convergence of including whether the sum converges absolutely or conditionally.

solution Discuss the convergence of including whether the sum converges absolutely or conditionally.

Ratio Test

For the infinite series ,

1. If , then the series converges absolutely.

2. If , then the series diverges.

3. If , then the test is inconclusive.

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

solution Discuss the convergence or divergence of the series .

Root Test

Let and

1. If , then converges absolutely.

2. If , then diverges.

3. If , this test is inconclusive.

solutionDiscuss the convergence or divergence of the series .

solutionDiscuss the convergence or divergence of the series .

solutionDiscuss the convergence or divergence of the series .

solutionDiscuss the convergence or divergence of the series .

solutionDiscuss the convergence or divergence of the series .

solutionDiscuss the convergence or divergence of the series .

solutionDiscuss the convergence or divergence of the series .

solutionDiscuss the convergence or divergence of the series .

Cauchy Condensation Test

The series and converge or diverge together.

solutionDiscuss the convergence or divergence of the series .

solutionDiscuss the convergence or divergence of the series .

solutionDiscuss the convergence or divergence of the series .

solutionDiscuss the convergence or divergence of the series .

solutionDiscuss the convergence or divergence of the series , where n is some real number.

Logarithmic Test

Raabe's Test

Series of Real Functions

solution Find the infinite series expansion of

solution Investigate the convergence of this series:

solution Investigate the convergence of this series:

solution Find the upper limit of the sequence

solution Find the upper limit of the sequence

solution Evaluate

solution Evaluate .

solution Find the upper limit of the sequence

solution Find the upper limit of the sequence

solution Evaluate

solution Determine the interval of convergence for the power series


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