Distance from a plane to an ellipsoid (Adapted from 1938

Explain how to approximate a function f at a point near (a, b) where the values of f, \(f_{x}, \text { and } f_{y}\), are known at (a, b).

Write the differential dw for the function w = f(x, y, z).

Write an equation for the plane tangent to the surface z = f(x,y) at the point (a, b, f(a, b)).

Differentials with more than two variables Write the differential dw in terms of the differentials of the independent variables. w=f(p, q, r, s)=p q /(r s)

Linear approximation a. Find the linear approximation for the following functions at the given point b. Use part (a) to estimate the given function value \(f(x, y)=\ln (1+x+y) ;(0,0) ; \text { estimate } f(0.1,-0.2) \text {. }\)

Tangent planes for z = f(x,y) Find an equation of the plane tangent to the following surface at the given points \(z=4-2 x^{2}-y^{2} ;(2,2,-8) \text { and }(-1,-1,1)\)

Differentials with more than two variables Write the differential dw in terms of the differentials of the independent variables. w=f(u, x, y, z)=(u+x) /(y+z)

Horizontal tangent planes Find the points at which the following surfaces have horizontal tangent planes. \(z=\cos 2 x \sin y \text { in the region }-\pi \leq x \leq \pi,-\pi \leq y \leq \pi\)

Tangent planes Find an equation of the plane tangent to the following surfaces at the given point (x+z) /(y-z)=2 ;(4,2,0)

Distance from a plane to an ellipsoid (Adapted from 1938 Putnam Exam.) Consider the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1\) and the plane P given by Ax + By + C2 + 1 = 0. Let \(h=\left(A^{2}+B^{2}+C^{2}\right)^{-1 / 2}) and\(m=\left(a^{2} A^{2}+b^{2} B^{2}+c^{2} C^{2}\right)^{1 / 2} .\) a. Find the equation of the plane tangent to the ellipsoid at the point (p, q, r). b. Find the two points on the ellipsoid at which the tangent plane is parallel to P and find equations of the tangent planes. c. Show that the distance between the origin and the plane P is h.

Horizontal tangent planes Find the points at which the following surfaces have horizontal tangent planes. \(x^{2}+y^{2}-z^{2}-2 x+2 y+3=0\)

Write the approximate change formula for a function z = f(x, y) at the point (a, b) in terms of differentials.

Tangent planes for F(x,y,z) = 0 Find an equation of the plane tangent to the following surfaces at the given points. \(z^{2}-x^{2} / 16-y^{2} / 9-1=0 ;(4,3,-\sqrt{3}) \text { and }(-8,9, \sqrt{14})\)

Linear approximation a. Find the linear approximation for the following functions at the given point b. Use part (a) to estimate the given function value f(x, y)=x y+x-y ;(2,3) estimate f(2.1,2.99) .

Tangent planes Find an equation of the plane tangent to the following surfaces at the given point \(\sin x y z=\frac{1}{2} ;\left(\pi, 1, \frac{1}{6}\right)\)

Approximate function change Use differentials to approximate the change in z for the given changes in the independent variables. \(z=-x^{2}+3 y^{2}+2\) when (x, y) changes from (-1, 2) to (-1.05, 1.9)

Tangent planes for F(x,y,z) = 0 Find an equation of the plane tangent to the following surfaces at the given points. x y+x z+y z-12=0 ;(2,2,2) and \(\left(-1,-2,-\frac{10}{3}\right)\)

Tangent planes for F(x,y,z) = 0 Find an equation of the plane tangent to the following surfaces at the given points. \(2 x+y^{2}-z^{2}=0 ;(0,1,1) \text { and }(4,1,-3)\)

Write an equation for the plane tangent to the surface F(x, y, z) = 0 at the point (a,b,c).

Approximate function change Use differentials to approximate the change in z for the given changes in the independent variables. z = In (1 + x + y) when (x, y) changes from (0,0) to (-0.1, 0.03)

Changes in cone volume The volume of a right circular cone with radius r and height h is \(V=\pi r^{2} h / 3\). a. Approximate the change in the volume of the cone when the radius changes from r = 6.5 to r = 6.6 and the height changes from h = 4.20 to h = 4.15. b. Approximate the change in the volume of the cone when the radius changes from r = 5.40 to r = 5.37 and the height changes from h = 12.0 to h = 11.96.

Suppose n is a vector normal to the tangent plane of the surface F(x, y, z) = 0 at a point. How is n related to the gradient of F at that point?

Tangent planes for z = f(x,y) Find an equation of the plane tangent to the following surface at the given points \(z=2 \cos (x-y)+2 ;(\pi / 6,-\pi / 6,3) \text { and }(\pi / 3, \pi / 3,4)\)

Write the explicit function \(z=x y^{2}+x^{2} y-10\) in the implicit form F(x, y, z) = 0.

Linear approximation a. Find the linear approximation for the following functions at the given point b. Use part (a) to estimate the given function value \(f(x, y)=(x+y) /(x-y) ;(3,2) ; \text { estimate } f(2.95,2.05) \text {. }\)

Tangent planes Find an equation of the plane tangent to the following surfaces at the given point \(z=\tan ^{-1}(x y) ;(1,1, \pi / 4)\)

Changes in torus surface area The surface area of a torus (an ideal bagel or doughnut) with an inner radius r and an outer radius \(R>r \text { is } S=4 \pi^{2}\left(R^{2}-r^{2}\right)\). a. If r increases and R decreases, does S increase or decrease, or is it impossible to say? b. If r increases and R increases, does S increase or decrease, or is it impossible to say? c. Estimate the change in the surface area of the torus when r changes from r = 3.00 to r = 3.05 and R changes from R = 5.50 to R = 5.65. d. Estimate the change in the surface area of the torus when r changes from r = 3.00 to r = 2.95 and R changes from R = 7.00 to R = 7.04. e. Find the relationship between the changes in r and R that leaves the surface area (approximately) unchanged.

Approximate function change Use differentials to approximate the change in z for the given changes in the independent variables. z = 2x – 3y - 2xy when (x, y) changes from (1, 4) to (1.1, 3.9)

Tangent planes for z = f(x,y) Find an equation of the plane tangent to the following surface at the given points \(z=(x-y) /\left(x^{2}+y^{2}\right)\); \(\left(1,2,-\frac{1}{5}\right)\) and \(\left(2,-1, \frac{3}{5}\right)\)

Area of an ellipse The area of an ellipse with axes of length 2a and 2b is A = \(\pi\)ab. Approximate the percent change in the area when a increases by 2% and b increases by 1.5%.

Horizontal tangent planes Find the points at which the following surfaces have horizontal tangent planes. \(x^{2}+2 y^{2}+z^{2}-2 x-2 z-2=0\)

Approximate function change Use differentials to approximate the change in z for the given changes in the independent variables. \(z=e^{x+y}\) when (x, y) changes from (0,0) to (0.1, -0.05)

Tangent planes for F(x,y,z) = 0 Find an equation of the plane tangent to the following surfaces at the given points. \(x^{2}+y^{2}-z^{2}=0\); (3,4,5) and (-4,-3,5)

Floating-point operations In general, real numbers (with infinite decimal expansions) cannot be represented exactly in a computer by floating-point numbers (with finite decimal expansions). Suppose that floating-point numbers on a particular computer carry an error of at most \(10^{-16}\). Estimate the maximum error that is committed in doing the following arithmetic operations. Express the error in absolute and relative (percent) terms. a. f(x, y) = xy b. f(x, y) = x/y c. F(x, y, z) = xyz d. F(x, y, z) = (x/y)/2

Linear approximation a. Find the linear approximation for the following functions at the given point b. Use part (a) to estimate the given function value \(f(x, y)=\sqrt{x^{2}+y^{2}} ;(3,-4) ; \text { estimate } f(3.06,-3.92) .\)

Probability of at least one encounter Suppose that in a large group of people a fraction \(0 \leq r \leq 1\) of the people have flu. The probability that in n random encounters, you will meet at least one person with flu is \(P=f(n, r)=1-(1-r)^{n}\). Although n is a positive integer, regard it as a positive real number. a. Compute \(f_{r} \text { and } f_{n}\) b. How sensitive is the probability P to the flu rate r? Suppose you meet n = 20 people. Approximately how much does the probability P increase if the flu rate increases from r = 0.1 to r = 0.11 (with n fixed)? c. Approximately how much does the probability P increase if the flu rate increases from r = 0.9 to r = 0.91? d. Interpret the results of parts (b) and (c).

Tangent planes for F(x,y,z) = 0 Find an equation of the plane tangent to the following surfaces at the given points. \(x y \sin z=1 ;(1,2, \pi / 6) \text { and }(-2,-1,5 \pi / 6)\)

Linear approximation a. Find the linear approximation for the following functions at the given point b. Use part (a) to estimate the given function value \(f(x, y)=-x^{2}+2 y^{2} ;(3,-1) ; \text { estimate } f(3.1,-1.04) \text {. }\)

Tangent planes for F(x,y,z) = 0 Find an equation of the plane tangent to the following surfaces at the given points. \(y z e^{x z}-8=0 ;(0,2,4) \text { and }(0,-8,-1)\)

Horizontal tangent planes Find the points at which the following surfaces have horizontal tangent planes. \(z=\sin (x-y) \text { in the region }-2 \pi \leq x \leq 2 \pi,-2 \pi \leq y \leq 2 \pi\)

Flow in a cylinder Poiseuille's Law is a fundamental law of fluid dynamics that describes the flow velocity of a viscous incompressible fluid in a cylinder (it is used to model blood flow through veins and arteries). It says that in a cylinder of radius R and length L, the velocity of the fluid r \(\leq\) R units from the centerline of the cylinder is \(V=\frac{P}{4 L \nu}\left(R^{2}-r^{2}\right)\), where P is the difference in the pressure between the ends of the cylinder and vis the viscosity of the fluid (see figure). Assuming that P and v are constant, the velocity V along the centerline of the cylinder (r = 0) is V = k\(R^{2}\)/L, where k is a constant that we will take to be k = 1. a. Estimate the change in the centerline velocity (r = 0) if the radius of the flow cylinder increases from R = 3 cm to R = 3.05 cm and the length increases from L = 50 cm to L = 50.5 cm. b. Estimate the percent change in the centerline velocity if the radius of the flow cylinder R decreases by 1% and the length L increases by 2%. c. Complete the following sentence: If the radius of the cylinder increases by p%, then the length of the cylinder must decrease by approximately ___% in order for the velocity to remain constant.

Linear approximation a. Find the linear approximation for the following functions at the given point b. Use part (a) to estimate the given function value \(f(x, y)=12-4 x^{2}-8 y^{2}\), (-1,4) estimate f(-1.05,3.95)

Law of cosines The side lengths of any triangle are related by the law of cosines. \(c^{2}=a^{2}+b^{2}-2 a b \cos \theta\) a. Estimate the change in the side length c when a changes from a = 2 to a = 2.03, b changes from b = 4.00 to b = 3.96 and \(\theta\) changes from \(\theta=\pi / 3 \text { to } \theta=\pi / 3+\pi / 90\) b. If a changes from a = 2 to a = 2.03 and b changes from b = 4.00 to b = 3.96, is the resulting change in c greater in magnitude when\(\theta=\pi / 20\) (small angle) or when \(\theta=9\pi / 20\) (close to a right angle)?

Explain how to approximate the change in a function f when the independent variables change from (a, b) to \((a+\Delta x, b+\Delta y)\).

Tangent planes for z = f(x,y) Find an equation of the plane tangent to the following surface at the given points \(z=2+2 x^{2}+y^{2} / 2\); \(\left(-\frac{1}{2}, 1,3\right)\) and (3,-2,22)

Tangent planes for z = f(x,y) Find an equation of the plane tangent to the following surface at the given points \(z=x^{2} e^{x-y} ;(2,2,4) \text { and }(-1,-1,1)\)

Tangent planes for z = f(x,y) Find an equation of the plane tangent to the following surface at the given points \(z=\ln (1+x y) ;(1,2, \ln 3) \text { and }(-2,-1, \ln 3)\)

Volume of a paraboloid The volume of a segment of a circular paraboloid (see figure) with radius r and height his \(V=\pi r^{2} h / 2\) Approximate the percent change in the volume when the radius decreases by 1.5% and the height increases by 2.2%.

Differentials with more than two variables Write the differential dw in terms of the differentials of the independent variables. \(w=f(x, y, z)=x y^{2}+z x^{2}+y z^{2}\)

Differentials with more than two variables Write the differential dw in terms of the differentials of the independent variables. w=f(x, y, z)=sin (x+y-z)

Travel cost The cost of a trip that is L miles long, driving a car that gets m miles per gallon, with gas costs of $p/gal is C = Lp/m dollars. Suppose you plan a trip of L = 1500 mi in a car that gets m = 32 mi/gal, with gas costs of p = $3.80/gal. a. Explain how the cost function is derived. b. Compute the partial derivatives \(C_{L}, C_{m}, \text { and } C_{p}\). Explain the meaning of the signs of the derivatives in the context of this problem. c. Estimate the change in the total cost of the trip if L changes from L = 1500 to L = 1520, m changes from m = 32 to 31, and p changes from $3.80 to p = $3.85. d. Is the total cost of the trip (with L = 1500 mi, m = 32 mi/gal, and p = $3.80) more sensitive to a 1% change in L, m, or p (assuming the other two variables are fixed)? Explain.

Explain why or why not Determine whether the following state ments are true and give an explanation or counterexample. a. The planes tangent to the cylinder \(x^{2}+z^{2}=1 \text { in } \mathbf{R}^{3}\) all have the form ax + bz + c = 0. b. Suppose w = xy/z for x > 0, y > 0, and 2 > 0. A decrease in z with x and y fixed results in an increase in w. c. The gradient \(\nabla F(a, b, c)\) lies in the plane tangent to the surface F(x, y, z) = 0 at (a, b,c).

Tangent planes Find an equation of the plane tangent to the following surfaces at the given point \(z=\tan ^{-1}(x+y) ;(0,0,0)\)

Heron's formula The area of a triangle with sides of length a, b, and c is given by a formula from antiquity called Heron's formula: \(A=\sqrt{s(s-a)(s-b)(s-c)}\) where s = (a + b + c)/2 is the semiperimeter of the triangle. a. Find the partial derivatives \(A_{a}, A_{b}, \text { and } A_{c}\) b. A triangle has sides of length a = 2, b = 4,c = 5. Estimate the change in the area when a increases by 0.03, b decreases by 0.08, and c increases by 0.6. c. For an equilateral triangle with a = b = c, estimate the percent change in the area when all sides increase in length by p%.

Surface area of a cone A cone with heighth and radius r has a lateral surface area (the curved surface only, excluding the base) of\(S=\pi r \sqrt{r^{2}+h^{2}}\) a. Estimate the change in the surface area when r increases from r = 2.50 to r = 2.55 and h decreases from h = 0.60 to h = 0.58 b. When r = 100 and h = 200, is the surface area more sensitive to a small change in r or a small change in h? Explain.

Line tangent to an intersection curve Consider the paraboloid \(z=x^{2}+3 y^{2}\) and the plane z = x + y + 4, which intersects the paraboloid in a curve Cat (2, 1, 7) (see figure). Find the equation of the line tangent to C at the point (2, 1, 7). Proceed as follows. a. Find a vector normal to the plane at (2, 1, 7). b. Find a vector normal to the plane tangent to the paraboloid at (2, 1, 7). c. Argue that the line tangent to Cat (2, 1, 7) is orthogonal to both normal vectors found in parts (a) and (b). Use this fact to find a direction vector for the tangent line. d. Knowing a point on the tangent line and the direction of the tangent line, write an equation of the tangent line in parametric form.

Batting averages Batting averages in baseball are defined by A = x/y, where x \(\geq\) 0 is the total number of hits and y> 0 is the total number of at-bats. Treat x and y as positive real numbers and note that \(0 \leq A \leq 1\). a. Use differentials to estimate the change in the batting average if the number of hits increases from 60 to 62 and the number of at-bats increases from 175 to 180. b. If a batter currently has a batting average of A = 0.350, does the average decrease if the batter fails to get a hit more than it increases if the batter gets a hit? c. Does the answer to part (b) depend on the current batting aver age? Explain.

Water-level changes A conical tank with radius 0.50 m and height 2.00 m is filled with water (see figure). Water is released from the tank, and the water level drops by 0.05 m (from 2.00 m to 1.95 m). Approximate the change in the volume of water in the tank. (Hint: When the water level drops, both the radius and height of the cone of water change.)

Two electrical resistors When two electrical resistors with resistance \(R_{1}\) > 0 and \(R_{2}\) > 0 are wired in parallel in a circuit (see figure), the combined resistance R is given by \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\) a. Estimate the change in Rif \(R_{1}\) increases from 2 ohms to 2.05 ohms and \(R_{2}\) decreases from 3 ohms to 2.95 ohms. b. Is it true that if \(R_{1}\) = \(R_{2}\) and \(R_{1}\) increases by the same small Explain. c. Is it true that if \(R_{1}\) and \(R_{2}\) increase, then R increases? Explain. d. Suppose \(R_{1}\) > \(R_{2}\), and \(R_{1}\) increases by the same small amount as \(R_{2}\) decreases. Does R increase or decrease?

Three electrical resistors Extending Exercise 58, when three electrical resistors with resistance \(R_{1}>0, R_{2}>0, \text { and } R_{3}>0) are wired in parallel in a circuit (see figure), the combined resistance R is given by \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}\). Estimate the change in R if \(R_{1}\) increases from 2 ohms to 2.05 ohms, \(R_{2}\) decreases from 3 ohms to 2.95 ohms, and \(R_{3}\) increases from 1.5 ohms to 1.55 ohms.

Power functions and percent change Suppose that \(z=f(x, y)=x^{a} y^{b}\), where a and b are real numbers. Let dx/x, dy/y, and dz/z be the approximate relative (percent) changes in x, y, and z, respectively. Show that (dz)/2 = aldx)/x + b(dy)/y; that is, the relative changes are additive when weighted by the exponents a and b.

Logarithmic differentials Let f be a differentiable function of one or more variables that is positive on its domain. a. Show that \(d(\ln f)=\frac{d f}{f}) b. Use part (a) to explain the statement that the absolute change in In f is approximately equal to the relative change in f. c. Let f(x, y) = xy, note that in f = In x + In y, and show that relative changes add; that is, (df) f = (dx)/x + (dy)/y. d. Let f(x, y) = x/y, note that in f = In x - In y, and show that relative changes subtract; that is, (df)/f = (dx)/x - (dy)/y. e. Show that in a product of n numbers, \(f=x_{1} x_{2} \cdots x_{n}\) the relative change in f is approximately equal to the sum of the relative changes in the variables.