Math Notes

Differential Calculus

Limits, continuity, derivatives, maxima/minima, and real applications (rates, optimization).

Solved Problems

Maxima - Minima

A box is to be constructed from a piece of zinc 20 inches square by cutting equal squares from each corner and burning up the zinc to form the side. What is the volume of the largest box that can be constructed?The number of newspaper copies distributed is given by C = 50 t<sup>2</sup> - 200t +10 00, where t is in years. What is the minimum number of copies distributed from 1995 to 2002?A cylinder is inscribed in a given sphere of radius a. Find the dimension of the cylinder if its lateral surface area is maximum.What should be the length of the third side of an isosceles triangle, if the congruent sides is 10 and the area is at maximum?The sum of two numbers is 15. What are these numbers if their product is as large as possible

Limits

Evaluate the limit: \( \lim_{\infty}(1+\frac{1}{n})(7-\frac{1}{n^2})\)Evaluate the limit: \( \lim_{x \to 1} (2-x)^{\tan {\frac{\pi x}{2}}} \)Limits

velocity problems

If s is measured in feet and t in seconds, what is the rate at which s is changing at the end of 2 seconds when s = 1/(1-t^2)Find the velocity of a point whose position is given by the equation \( s = 14t^3 + 5t -1\), where t = 2.

Slopes

Find the slope of the curve y = (x+1)(x+2) at points where it crosses the x-axis. Trace the curve.At what point does the curve y = x/(x+1), have the slope 1/4?Find the slope of the curve \(x^2 +xy +y^2 =3 \) and \( (1, 1)\)Find the slopes of the curve \(s^2 +y^2 = 25\) at (3,4).Find the slope of the tangent line to the curve given by the equation \( y = 3x^2 +2x+ 1\) at x =2.

Time Rates

A standard cell has an emf \(E\) of \(1.2\) volts. If the resistance \(R\) of the circuit is increasing at the rate of 0.03 ohm/sec, at what rate is the current (I) changing at the instant when the resistance is 6 ohms? Assume Ohm's Law E=IR A 5m long ladder leans against a vertical wall 4m high. If the lower end is sliding at 1 m/sec, how fast is the tip of the ladder moving?A triangular trough is 10ft long 4ft across the top and 4 feet deep. If water flows in at the rate of 30 cu. ft. per min., find how fast the surface rising when the water is 6 in. deep.A rectangular trough is 10 ft long and 3 ft wide. Find how fast the surface rises, if water flows in at the rate of 12 cu. ft. per min.Water flows into a vertical cylindrical tank at 12 cu. ft. per min.; the surface rises at 6 in. per min. Find the radius of the tank.Water is flowing into a vertical cylindrical tank at the rate of 24 cu. ft. per min. If the radius of the tank is 4ft. how fast is the surface rising?The distance of y from the origin at time t is given by y =16t<sup>2</sup> + 3000 t +50, 000. What is the initial velocity when t = 0?

Geometric Applications

Find the tangents to the curve \(y = \frac{1}{2} x^4 -x^3 +5x \) which makes an angle of \(45^o\) with the x-axis.Find the angle between parabolas \(y^2 =x\) and \(y =x^2\) at the points of their intersections.Find the tangent and normal to the curve \(x^2 -2xy +2y^2 -x = 0\) at \(x=1\).

Derivatives

What is the first derivative of the functions, y = x/ sqrt (a^2 -x^2 )^3

Tags:

differential calculus limits continuity derivatives chain rule maxima and minima related rates optimization