Chapter 17

The equation \(x^{2}+y^{2}=169\) describes a circle with radius 13 centered at the origin. (a) Solve explicitly for \(y\) in terms of \(x\). Is \(y\) a function of \(x\) ? (b) Differentiate your expression(s) from part (a) to find \(\frac{d y}{d x}\). (c) Now use implicit differentiation on the original equation to find \(\frac{d y}{d x}\). (d) Which method of differentiation (that used in part(a), or that used in part (b)) was easier? Why? (e) What is the slope of the tangent line to the circle at the point \((5,12) ?\) At \((5,-12)\) ?

Differentiate the following. (a) \(y=3^{x}\) (b) \(y=x^{3}\) (c) \(y=x^{x}\), where \(x>0\).

Find \(f^{\prime}(x)\). (a) \(f(x)=2 x^{x}\), where \(x>0\) (b) \(f(x)=5\left(x^{2}+1\right)^{x}\) (c) \(f(x)=\left(2 x^{4}+5\right)^{3 x+1}\)

Suppose we toss a rock into a pond causing a circular ripple. If the radius is increasing at a rate of 3 feet per second when the diameter is 4 , how fast is the area increasing?

Which is larger, the slope of the tangent line to \(x^{2}+y^{2}=25\) at the point \((4,-3)\) or the slope of the tangent line to \(x^{2}+4 y^{2}=25\) at the point \((4,-3 / 2) ?\) You can answer this analytically (find the two slopes), or you can answer this by looking at the graphs of the ellipse and the circle and sketching the tangent to each at the designated points. (To sketch the ellipse, look at the \(x\) -and \(y\) -intercepts.)

Differentiate the following. $$ y=(x+1)^{(x+1)}, \text { where } x>-1 \text { . } $$

Find \(f^{\prime}(x)\). (a) \(f(x)=3 \cdot 2^{x}+2 \cdot x^{3}+3 \cdot x^{2 x+3}\), where \(x>0\) (b) \(f(x)=x\left(2 x^{3}+1\right)^{x}+5\), where \(x>0\)

The price and the demand for a certain item can be modeled by the equation \(20 p=-q+200\) (a) Express the rate of change of quantity demanded with respect to price in terms of a derivative and evaluate it. (b) Suppose that price is determined by the world market, so that price and quantity can both be thought of as functions of time. At a certain instant the price is \(\$ 6\) and is increasing at a rate of \(\$ 0.25\) per week. At what rate is the quantity demanded changing at this instant? Why is your answer negative?

Consider the equation \(x^{2} y+x y^{2}+x=1\). Find \(\frac{d y}{d x}\) at all points where \(x=1\).

Differentiate the following. $$ y=\left(3 x^{2}+2\right)^{x} $$

Find \(g^{\prime}(t)\) (a) \(g(t)=\frac{2^{t}}{t^{2 t}}\), where \(t>0\) (b) \(g(t)=\ln (t+1)^{t^{2}+1}\), where \(t>-1\)

A bug is walking around the circle \(x^{2}+y^{2}=169 .\) At a certain instant the bug is at the point \((-5,12)\) and its \(y\) -coordinate is decreasing at a rate of 3 units per second. (a) Is the bug traveling the circle in a clockwise direction or a counterclockwise direction? (b) How fast is its \(x\) -coordinate changing at this instant?

Find the slopes of the tangent lines to \((x-2)^{2}+(y-3)^{2}=25\) at the two points where \(x=6\).

Differentiate the following. $$ y=x^{x^{2}}, \text { where } x>0 $$

Find \(\frac{d y}{d x}\) using logarithmic differentiation. You need not simplify. (a) \(y=x^{\ln \sqrt{x}}\), where \(x>0\) (b) \(y=\frac{x e^{5 x}}{(x+1)^{2} \sqrt{x-2}}\), where \(x>0\) (c) \(y=\left(e^{2 x}\right)\left(x^{2}+3\right)^{5}\left(2 x^{2}+1\right)^{3}\)

A spherical balloon is losing air at a steady rate of \(0.5 \mathrm{~cm}^{3} /\) hour. (a) How fast is the radius decreasing when the diameter of the balloon is \(30 \mathrm{~cm}\) ? (b) How fast is the radius decreasing when the diameter of the balloon is \(15 \mathrm{~cm}\) ? (c) How fast is the diameter decreasing when the diameter of the balloon is \(15 \mathrm{~cm}\) ? (d) How fast is the radius decreasing when the circumference of the balloon is \(12 \pi\) centimeters?

Find the equation of the line tangent to the curve \(x^{3}+y^{3}-3 x^{2} y^{2}+1=0\) at the point \((1,1) .\)

Suppose \(y=f(x) g(x)\), where \(f(x)\) and \(g(x)\) are positive for all \(x\). Use logarithmic differentiation to find \(\frac{d y}{d x}\). Verify that your result is simply the Product Rule.

A cylindrical barrel with radius 3 feet is lled with oil. The barrel stands upright and oil comes out a faucet at the base of the tank. (a) How are the volume of oil in the tank and the height of oil in the tank related? (b) What is the rate of change of volume with respect to height? Is it constant, or does it depend upon the height? Does your answer make sense to you? (c) As oil leaves the tank, both the volume and the height of oil in the tank change with respect to time. i. When the height is decreasing at a rate of \(0.5\) feet per hour, how fast is the volume of oil in the tank changing? ii. When oil is leaving the tank at a rate of 3 cubic feet per hour, how fast is the height of oil in the tank changing?

Find \(\frac{d y}{d x}\) for the curve \(x^{3}+3 y+y^{2}=6\). What is the slope at \((2,-1)\) ? At what points is the slope zero?

Suppose \(y=\frac{f(x)}{g(x)}\), where \(f(x)\) and \(g(x)\) are positive for all \(x\). Use logarithmic differentiation to find \(\frac{d y}{d x}\). Verify that this is the same result you would get had you used the Quotient Rule.

At 7:00 A.M. a truck is 100 miles due north of a car. The truck is traveling south at a constant speed of \(40 \mathrm{mph}\), while the car is traveling east at \(60 \mathrm{mph}\). How fast is the distance between the car and the truck changing at 7:30 A.M.?

At what points (if any) is the tangent line to the curve \(3 x^{2}+6 x y+8 y^{2}=8\) vertical?

As shown on the following page, a wheelbarrow is being wheeled down a perfectly straight 13 -foot ramp. The top of the ramp is 5 feet high and the ramp covers a horizontal distance of 12 feet. When the wheelbarrow has moved a horizontal distance of 6 feet it is moving horizontally at a rate of \(0.5\) feet per second. At what rate is it moving vertically downward?

(a) Graph the ellipse \(4(x-1)^{2}+9(y-3)^{2}=36 .\) (Sketch \(4 x^{2}+9 y^{2}=36\) by looking for \(x\) - and \(y\) -intercepts and then shift the graph horizontally and vertically as appropriate.) (b) Find the slope of the line tangent to the ellipse at the point \((1,5)\) in two ways. First solve for \(y\) explicitly (using the appropriate half of the ellipse) and find \(\frac{d y}{d x}\). Then do the problem by differentiating implicitly.

Valentina is sipping from a straw she has stuck in a conical cup of lemonade. The cup is an inverted circular cone of height \(12 \mathrm{~cm}\) and radius \(6 \mathrm{~cm}\). When the height of the lemonade in the cup is \(10 \mathrm{~cm}\), Valentina is sipping lemonade at a rate of \(2 \mathrm{~cm}^{3} / \mathrm{second}\). At this moment, how fast is the height of the lemonade in the cup decreasing?

Consider the circle given by \(x^{2}+y^{2}=4\). (a) Show that \(\frac{d y}{d x}=-\frac{x}{y}\). (b) Show that \(\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(-\frac{x}{y}\right)=-\frac{4}{y^{3}} .\) Explain your reasoning completely.

Two vehicles are on parallel roads that are \(0.5\) miles apart. At noon the vehicles are half a mile apart and traveling in the same direction. One is traveling at \(30 \mathrm{mph}\) and the other at \(40 \mathrm{mph}\). (a) After 2 hours, how fast is the distance between the cars increasing? (b) When the cars are 40 miles apart, how fast is the distance between them increasing? (c) What would be the answer to part (a) if the cars were traveling in opposite directions?

The equation \(2\left(x^{2}+y^{2}\right)^{2}=25\left(x^{2}-y^{2}\right)\) gives a curve that is known as a lemniscate. Find the slope of the tangent line to the lemniscate at the point \((-3,1)\).

In Barcelona there is a beautiful Spanish castle set \(1 / 4\) of a mile back from a straight road. A bicyclist rides by the castle at a velocity of \(15 \mathrm{mph}\). Assuming that the biker maintains this speed, how fast is the distance between the biker and the castle increasing 20 minutes later?

Find an equation for the tangent line to \(y^{2}=x^{3}(3-x)\) at the point \((1,2)\). What can you say about the tangent line to this curve at the point \((3,0)\) ?

A conical container is used to hold oil. It is positioned upright with the tip of the cone at the bottom. Oil comes out a faucet at the base of the container. We know that the volume of a cone is \(V=\frac{1}{3} \pi r^{2} h\). (a) As oil leaves the cone the height, radius, and volume of oil in the container change with time. Find \(\frac{d V}{d t}\) in terms of \(r, h, \frac{d r}{d t}\), and \(\frac{d h}{d t}\). (b) Suppose the container has a height of 24 inches and a radius of 12 inches. Express the volume of oil in the container as a function of the height of oil in the container. (Hint: Use similar triangles to express the radius of the oil in terms of the height of the oil.) (c) Suppose oil is leaking out of the container at a rate of 5 cubic inches per hour. How fast is the height of the oil in the container decreasing when the height is 10 inches? When the height is 4 inches? Do the relative sizes of your answers make sense to you intuitively?

Find \(\frac{d y}{d x}\). (a) \(3 x^{2}+6 y^{2}+3 x y=10\) (b) \((x-2)^{3} \cdot(y-2)^{3}=1\) (c) \(x y^{2}+2 y=x^{2} y+1\) (d) \(\left(x^{2} y^{3}+y\right)^{2}=3 x\) (e) \(e^{x y}=y^{2}\) (f) \(x \ln \left(x y^{3}\right)=y^{2}\) (g) \(\ln (x y)=x y^{2}\)

An airplane is ying at a speed of \(600 \mathrm{mph}\) at a constant altitude of 1 mile. It passes directly over an air traf c control tower. How fast is the distance between the control tower and the airplane increasing when the plane is 10 miles away from the tower?

(a) Sketch a graph of \((x-2)^{2}-(y-3)^{2}=25\). (Sketch \(x^{2}-y^{2}=25\) by looking for \(x\) - and \(y\) -intercepts and then shift your graph vertically and horizontally as appropriate.) (b) Find the equation of the tangent line to the graph at each of the following points. i. \((15,-9)\) ii. \((-3,3)\)

Sand is being dumped into a conical pile whose height is \(1 / 2\) the radius of its base. Suppose sand is being dumped at a rate of 5 cubic meters per minute. (a) How fast is the height of the pile increasing when it is 9 meters high? (b) How fast is the area of the base increasing at this moment? (c) How fast is the circumference of the base increasing at this moment? (d) Will the height be increasing more slowly, more rapidly, or at a steady pace as time goes on?

Find an equation for the tangent line to \(x^{2 / 3}+y^{2 / 3}=5\) at the point \((8,1)\).