Question: Find the limit. Use l’Hospital’s
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Describe how the graph of f varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also
> The graph shows the fuel consumption of a car (measured in gallons per hour) as a function of the speed of the car. At very low speeds the engine runs inefficiently, so initially decreases as the speed increases. But at high speeds the fuel consumption i
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f" to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = x – 1
> Describe how the graph of f varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also
> Describe how the graph of f varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also
> Describe how the graph of f varies as varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also
> The family of functions f (t) = C (e-at – e-bt), where a, b, and C are positive numbers and b > a, has been used to model the concentration of a drug injected into the bloodstream at time t = 0. Graph several members of this family. What do they have in
> Investigate the family of curves given by the parametric equations x= t3 - ct, y =t2. In particular, determine the values of for which there is a loop and find the point where the curve intersects itself. What happens to the loop as c increases? Find the
> Graph the curve in a viewing rectangle that displays all the important aspects of the curve. At what points does the curve have vertical or horizontal tangents? x = t* + 41 – 81?, y= 2t2 – t
> Graph the curve in a viewing rectangle that displays all the important aspects of the curve. At what points does the curve have vertical or horizontal tangents? x = t* – 2t° – 2t', y=r°- t
> Use a graph to estimate the coordinates of the leftmost point on the curve x = t4 – t2, y = t + ln t. Then use calculus to find the exact coordinates.
> In Example 4 we considered a member of the family of functions f (x) = sin (x + sin cs) that occur in FM synthesis. Here we investigate the function with c = 3. Start by graphing f in the viewing rectangle [0, π] by [-1.2, 1.2]. How many local maximum po
> Graph f (x) = ex + ln |x – 4| using as many viewing rectangles as you need to depict the true nature of the function.
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f' and f" to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = x° –
> Use a computer algebra system to graph f and to find f' and f". Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of, f. 1- elz 1+ el f(x) = %3D
> Use a computer algebra system to graph f and to find f' and f". Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of, f. f(x) = (x² – 1)e' arctan x
> In this project we investigate the most economical shape for a can. We first interpret this to mean that the volume V of a cylindrical can is given and we need to find the height and radius that minimize the cost of the metal to make the can (see the fig
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> If f' is continuous, f (2) = 0, and f'(2) = 7, evaluate limx→0 f (2 + 3x) + f (2 + 5x)/x
> Suppose is a positive function. If limx→a f (x) = 0 and limx→a g (x) = ∞, show that limx→a [f (x)] g (x) = 0. This shows that is not an indeterminate form.
> Evaluate limx→∞ [x - x2 ln (1 + x/x)].
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Use a computer algebra system to graph f and to find f' and f". Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of, f. f(x) = x+ 5 sin x, x< 20
> The first appearance in print of l’Hospital’s Rule was in the book Analyse des Infiniment Petits published by the Marquis de l’Hospital in 1696. This was the first calculus textbook ever published and
> A metal cable has radius and is covered by insulation, so that the distance from the center of the cable to the exterior of the insulation is R. The velocity v of an electrical impulse in the cable is where c is a positive constant. Find the following
> If an electrostatic field E acts on a liquid or a gaseous polar dielectric, the net dipole moment P per unit volume is Show that lim E→0+ P (E) = 0. e + e-E P(E) eE – e-E E
> If an object with mass m is dropped from rest, one model for its speed after seconds, taking air resistance into account, is where g is the acceleration due to gravity and is a positive constant. (In Chapter 7 we will be able to deduce this equation from
> If an initial amount A0 of money is invested at an interest rate r compounded times a year, the value of the investment after t years is If we let n→∞, we refer to the continuous compounding of interest. Use lâ
> Prove that for any number p > 0. This shows that the logarithmic function approaches ∞ more slowly than any power of x. In x = 0 lim rP
> Prove that for any positive integer n. This shows that the exponential function approaches infinity faster than any power of x. lim = 00
> Investigate the family of curves f (x) = ex - cx. In particular, find the limits as x→±∞ and determine the values of for which has an absolute minimum. What happens to the minimum points as increases?
> What happens if you try to use l’Hospital’s Rule to evaluate Evaluate the limit using another method. lim x² + 1
> The blood vascular system consists of blood vessels (arteries, arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expended by the heart in pumping
> Use a computer algebra system to graph f and to find f' and f". Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of, f. 2/3 f(x) 1+x + x*
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> A painting in an art gallery has height and is hung so that its lower edge is a distance above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer sta
> Where should the point P be chosen on the line segment AB so as to maximize the angle θ? В 2 P 3 A
> A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle θ. How should θ be chosen so that the gutter will carry the maximum amount of water? -
> Use a computer algebra system to graph f and to find f' and f". Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of, f. f(x) = x? +x + 1
> Use l’Hospital’s Rule to help find the asymptotes of f. Then use them, together with information from f' and f", to sketch the graph of f. Check your work with a graphing device. f(x) = xe
> The upper right-hand corner of a piece of paper, 12 in. by 8 in., as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y? 12 - y 8
> Use l’Hospital’s Rule to help find the asymptotes of f. Then use them, together with information from f' and f", to sketch the graph of f. Check your work with a graphing device. f(x) = e"/x
> Let v1 be the velocity of light in air and v2 the velocity of light in water. According to Fermat’s Principle, a ray of light will travel from a point A in the air to a point B in the water by a path ACB that minimizes the time taken. S
> Illustrate l’Hospital’s Rule by graphing both f (x)/g (x) and f'(x)/ g'(x) near x = 0 to see that these ratios have the same limit as x→0. Also, calculate the exact value of the limit. f(x) = 2x
> Find the points on the ellipse 4x2 + y2 = 4 that are farthest away from the point (1, 0).
> Illustrate l’Hospital’s Rule by graphing both f (x)/g (x) and f'(x)/ g'(x) near x = 0 to see that these ratios have the same limit as x→0. Also, calculate the exact value of the limit. f(x) = e*
> Use a graph to estimate the value of the limit. Then use l’Hospital’s Rule to find the exact value. 5 - 4" lim 1-0 3* - 2*
> Use a graph to estimate the value of the limit. Then use l’Hospital’s Rule to find the exact value.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> What is the smallest possible area of the triangle that is cut off by the first quadrant and whose hypotenuse is tangent to the parabola y = 4 – x2 at some point?
> What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve y = 3/x at some point?
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> If is the function of Exercise 12, find f' and f" and use their graphs to estimate the intervals of increase and decrease and concavity of f. Exercise 12: Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sket
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10 f
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end as in the figure. It is believed that bees form their cells in such a way as to where s, the length of the sides of the hexagon, and h, the heig
> For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to v3. It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a
> If a resistor of R ohms is connected across a battery of E volts with internal resistance ohms, then the power (in watts) in the external resistor is If E and r, are fixed but R varies, what is the maximum value of the power?
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> (a). Show that of all the rectangles with a given area, the one with smallest perimeter is a square. (b). Show that of all the rectangles with a given perimeter, the one with greatest area is a square.
> A cone with height h is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h = 1/3H.
> A cone-shaped paper drinking cup is to be made to hold 27 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper.
> A cone-shaped drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup.
> A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?
> A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum? (b) A minimum?
> A right circular cylinder is inscribed in a cone with height and base radius r. Find the largest possible volume of such a cylinder.
> A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus, the diameter of the semicircle is equal to the width of the rectangle) If the perimeter of the window is 30 ft, find the dimensions of the window so that the greatest possibl
> A cylindrical can without a top is made to contain V cm3 of liquid. Find the dimensions that will minimize the cost of the metal to make the can.
> Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r.
> Find the area of the largest rectangle that can be inscribed in the ellipse x2/a2 + y2/b2 = 1.
> Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs (with a graphing device) that display the major features of the curve. Use these graphs to estimate the maximum and minimum
> A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible volume of such a cylinder.
> Find the dimensions of the rectangle of largest area that has its base on the -axis and its other two vertices above the x-axis and lying on the parabola y = 8 – x2.
> Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle.
> Find, correct to two decimal places, the coordinates of the point on the curve y = tan x that is closest to the point (1, 1).
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for t
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> A box with a square base and open top must have a volume of 32,000 cm3. Find the dimensions of the box that minimize the amount of material used.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> If 1200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> The rate (in mg carbon/m3/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function P = 100I/I2 + I + 4 where I is the light intensity (measured in thousands of foot candles). For what light intensity is P a maximum
> A model used for the yield of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) is Y = KN/1 + N2 where is a positive constant. What nitrogen level gives the best yield?
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Given that which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible.
> Find two positive numbers whose product is 100 and whose sum is a minimum.
> Find two numbers whose difference is 100 and whose product is a minimum.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why.
> Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. (a)
> Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum. (a). Make a table of values, like the following one, so that the sum of the numbers in the first two columns is always 23. On the basis of the evidence in yo
