Question: The DNA molecule has the shape of
The DNA molecule has the shape of a double helix (see Figure 3 on page 850). The radius of each helix is about 10 angstroms (1 Å = 10-8 cm). Each helix rises about 34 Å during each complete turn, and there are about 2.9 × 108 complete turns. Estimate the length of each helix.
Figure 3:
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> Find the distance between the given parallel planes. 2x - 3y + z = 4, 4x - 6y + 2z = 3
> Find the distance from the point to the given plane. (-6, 3, 5), x - 2y - 4z = 8
> Find the distance from the point to the given plane. (1, -2, 4), 3x + 2y + 6z = 5
> (a). Find symmetric equations for the line that passes through the point (1, -5, 6) and is parallel to the vector −1, 2, −3 . (b). Find the points in which the required line in part (a) intersects the coordinate planes.
> A particle moves according to a law of motion s = f(t), t ≥ 0, where t is measured in seconds and s in feet. (a) Find the velocity at time t. (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the part
> Use the formula in Exercise 12.4.45 to find the distance from the point to the given line. (0, 1, 3); x = 2t, y = 6 - 2t, z = 3 + t Exercise 12.4.45: x b| |a| Ja x b| |a X d =
> Find parametric equations and symmetric equations for the line. The line through the points (0, 1 2 , 1) and (2, 1, -3)
> Use the formula in Exercise 12.4.45 to find the distance from the point to the given line. (4, 1, -2); x = 1 + t, y = 3 - 2t, z = 4 - 3t Exercise 12.4.45: x b| |a| Ja x b| |a X d =
> Find parametric equations and symmetric equations for the line. The line through the origin and the point (4, 3, -1)
> Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) 5x + 2y + 3z = 2, y = 4x - 6z
> Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) 2x - 3y = z, 4x = 3 + 6y + 2z
> (a) Estimate the area under the graph of f (x) = 1 + x2 from x = -1 to x = 2 using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles. (b) Repeat part (a) using left
> Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) x - y + 3z = 1, 3x + y - z = 2
> A curve passes through the point (0, 5) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. What is the equation of the curve?
> (a) Estimate the area under the graph of f (x) = sin x from x = 0 to x = π/2 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using le
> (a) Estimate the area under the graph of f (x) = 1/x from x = 1 to x = 2 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left en
> (a) Use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 12. (i) L6 (sample points are left endpoints) (ii) R6 (sample points are right endpoints) (iii) M6 (sample points are midpoints) (b) Is L6 an
> (a) By reading values from the given graph of f, use five rectangles to find a lower estimate and an upper estimate for the area under the given graph of f from x = 0 to x = 10. In each case sketch the rectangles that you use. (b) Find new estimates usin
> Find the exact area under the cosine curve y = cos x from x = 0 to x = b, where 0 ≤ b ≤ π/2. (Use a computer algebra system both to evaluate the sum and compute the limit.) In particular, what is the area if b = π/2 ?
> Find the exact area of the region under the graph of y = e-x from 0 to 2 by using a computer algebra system to evaluate the sum and then the limit in Example 3(a). Compare your answer with the estimate obtained in Example 3(b). Example 3(a) & 3(b):
> (a) Express the area under the curve y = x5 from 0 to 2 as a limit. (b) Use a computer algebra system to find the sum in your expression from part (a). (c) Evaluate the limit in part (a).
> Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.) 9x - 3y + 6z = 2, 2y = 6x + 4z
> Dinosaur fossils are often dated by using an element other than carbon, such as potassium-40, that has a longer half-life (in this case, approximately 1.25 billion years). Suppose the minimum detectable amount is 0.1% and a dinosaur is dated with 40K to
> The table shows the number of people per day who died from SARS in Singapore at two-week intervals beginning on March 1, 2003. (a) By using an argument similar to that in Example 4, estimate the number of people who died of SARS in Singapore between Ma
> In someone infected with measles, the virus level N (measured in number of infected cells per mL of blood plasma) reaches a peak density at about t = 12 days (when a rash appears) and then decreases fairly rapidly as a result of immune response. The area
> Find the cosine of the angle between the planes x + y + z = 0 and x + 2y + 3z = 1.
> (a) Graph the function f (x) = x - 2 ln x 1 ≤ x ≤ 5 (b) Estimate the area under the graph of f using four approximating rectangles and taking the sample points to be (i) right endpoints and (ii) midpoints. In each case sketch the curve and the rectangles
> Find the length of the curve. r(t) = i + t2 j + t3 k, 0 < t < 1
> Find the length of the curve. r(t) = 2 t i + et j + e-t k, 0 < t < 1
> Find the length of the curve. r(t) = 〈21, t?, 불r3), 0<t<1
> Try to sketch by hand the curve of intersection of the parabolic cylinder y = x2 and the top half of the ellipsoid x2 + 4y2 + 4z2 = 16. Then find parametric equations for this curve and use these equations and a computer to graph the curve.
> Find, correct to four decimal places, the length of the curve of intersection of the cylinder 4x2 + y2 = 4 and the plane x + y + z = 2
> Graph the curve with parametric equations x = sin t, y = sin 2t, z = sin 3t. Find the total length of this curve correct to four decimal places.
> The rectifying plane of a curve at a point is the plane that contains the vectors T and B at that point. Find the rectifying plane of the curve r (t) = sin t i + cos t j + tan t k at the point ( 2 /2, 2 /2, 1).
> Find equations of the normal and osculating planes of the curve of intersection of the parabolic cylinders x = y2 and z = x2 at the point (1, 1, 1).
> Is there a point on the curve in Exercise 53 where the osculating plane is parallel to the plane x + y + z = 1? [Note: You will need a CAS for differentiating, for simplifying, and for computing a cross product.] Exercise 53: At what point on the curve
> At what point on the curve x = t 3, y = 3t, z = t4 is the normal plane parallel to the plane 6x + 6y - 8z = 1?
> Find equations of the osculating circles of the parabola y = 1 2 x2 at the points (0, 0) and (1, 1 2 ). Graph both osculating circles and the parabola on the same screen.
> Find equations of the osculating circles of the ellipse 9x2 + 4y2 = 36 at the points (2, 0) and (0, 3). Use a graphing calculator or computer to graph the ellipse and both osculating circles on the same screen.
> Find equations of the normal plane and osculating plane of the curve at the given point. x = ln t, y = 2t, z = t2; (0, 2, 1)
> Find the vectors T, N, and B, at the given point. r(t) = (cos t, sin t, In cos t), (1,0, 0)
> Consider the curvature at x = 0 for each member of the family of functions f (x) = ecx. For which members is k (0) largest?
> Two graphs, a and b, are shown. One is a curve y = f (x) and the other is the graph of its curvature function y = k (x). Identify each curve and explain your choices. y. a b
> Two graphs, a and b, are shown. One is a curve y = f (x) and the other is the graph of its curvature function y = k (x). Identify each curve and explain your choices. y b
> Plot the space curve and its curvature function k (t). Comment on how the curvature reflects the shape of the curve. r(t) = (te', e*, /Zt), -5<t<5
> Plot the space curve and its curvature function k (t). Comment on how the curvature reflects the shape of the curve. r(t) = (t – sin t, 1 – cos t, 4 cos(t/2)), 0<ts 87 %3D
> (a). Is the curvature of the curve C shown in the figure greater at P or at Q? Explain. (b). Estimate the curvature at P and at Q by sketching the osculating circles at those points. у. P
> At what point does the curve have maximum curvature? What happens to the curvature as x →∞ ? y = ln x
> Let’s consider the problem of designing a railroad track to make a smooth transition between sections of straight track. Existing track along the negative x-axis is to be joined smoothly to a track along the line y = 1 for x > 1. (a)
> Find the curvature and torsion of the curve x = sin h t, y = cos h t, z = t at the point (0, 1, 0).
> Graph the curve with parametric equations x = cos t, y = sin t, z = sin 5t and find the curvature at the point (1, 0, 0).
> Reparametrize the curve with respect to arc length measured from the point (1, 0) in the direction of increasing t. Express the reparametrization in its simplest form. What can you conclude about the curve? 2 2t r(t): i + t2 + 1 t2 + 1
> Show that the tangent vector to a curve defined by a vector function r(t) points in the direction of increasing t. [Hint: Refer to Figure 1 and consider the cases h > 0 and h Figure 1: r(t+h)– r(t) r(t+h) – r(t) h r'(t) P. /r(t) r(t+h) r(t) r(t+
> Prove Formula 5 of Theorem 3.
> Find r (t) if r' (t) = t i + e' t j + te' k and r (0) = i + j + k.
> Suppose u and v are vector functions that possess limits as t → a and let c be a constant. Prove the following properties of limits. (a) lim [u(t) + v(t)] = lim u(t) + lim v(t) a a (b) lim cu(t) = c lim u(t) a (c) lim [u(t) • v(t)]
> (a). Graph the curve with parametric equations (b). Show that the curve lies on the hyperboloid of one sheet 144x2 + 144y2 - 25z2 = 100. 27 x = sin 8t – sin 181 8 y = -% cos 8t + 39 cos 18t z = * sin 5t
> Two particles travel along the space curves Do the particles collide? Do their paths intersect? r, (1) = (t, t°, t³) r2(t) = (1 + 2t, 1 + 6t, 1 + 14t)
> If two objects travel through space along two different curves, it’s often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know w
> Try to sketch by hand the curve of intersection of the circular cylinder x2 + y2 = 4 and the parabolic cylinder z = x2. Then find parametric equations for this curve and use these equations and a computer to graph the curve.
> Find a vector function that represents the curve of intersection of the two surfaces. The semiellipsoid x2 + y2 + 4z2 = 4, y > 0, and the cylinder x2 + z2 = 1
> Find a vector function that represents the curve of intersection of the two surfaces. The hyperboloid z = x2 - y2 and the cylinder x2 + y2 = 1
> Find a vector function that represents the curve of intersection of the two surfaces. The paraboloid z = 4x2 + y2 and the parabolic cylinder y = x2
> Find a vector equation for the tangent line to the curve of intersection of the cylinders x2 + y2 = 25 and y2 + z2 = 20 at the point (3, 4, 2).
> Find a vector function that represents the curve of intersection of the two surfaces. The cylinder x2 + y2 = 4 and the surface z = xy
> Show that the curve with parametric equations x = t2, y = 1 - 3t, z = 1 + t3 passes through the points (1, 4, 0) and (9, -8, 28) but not through the point (4, 7, -6).
> Graph the curve with parametric equations Explain the appearance of the graph by showing that it lies on a sphere. x = V1 – 0.25 cos? 10t cos t y = v1 – 0.25 cos² 10t sin t z = 0.5 cos 10t
> Graph the curve with parametric equations x = (1 + cos 16t) cos t y = (1 + cos 16t) sin t z = 1 + cos 16t Explain the appearance of the graph by showing that it lies on a cone.
> Graph the curve with parametric equations x = sin t, y = sin 2t, z = cos 4t. Explain its shape by graphing its projections onto the three coordinate planes.
> Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = e-t cos t, y = e-t sin t, z = e-t; (1, 0, 1)
> Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = ln (t + 1), y = t cos 2t, z = 2t, (0, 0, 1)
> Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = t? + 1, y= 4/t, z = e“-'; (2,4, 1) %3D
> At what points does the curve r(t) = t i + (2t - t2) k intersect the paraboloid z = x2 + y2?
> Find three different surfaces that contain the curve r(t) = t2 i + ln t j + s(1/t) k.
> Find three different surfaces that contain the curve r(td) = 2t i + et j + e2t k.
> Show that the curve with parametric equations x = sin t, y = cos t, z = sin2t is the curve of intersection of the surfaces z = x2 and x2 + y2 = 1. Use this fact to help sketch the curve.
> Show that the curve with parametric equations x = t cos t, y = t sin t, z = t lies on the cone z2 = x2 + y2, and use this fact to help sketch the curve.
> Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices. x = cos2 t, y = sin2 t, z = t ZA
> Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices. x = cos 8t, y = sin 8t, z = e0.8t, t > 0 ZA
> Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices. x = cos t, y = sin t, z = cos 2t ZA
> Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices. x = t, y = 1 / (1 + t2), z = t2 ZA II
> Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices. x = cos t, y = sin t, z = 1/ (1 + t2) ZA -y
> Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices. x = t cos t, y = t, z = t sin t, t > 0 ZA
> (a). Find the arc length function for the curve measured from the point P in the direction of increasing t and then reparametrize the curve with respect to arc length starting from P, and (b). find the point 4 units along the curve (in the direction of
> (a). Find the arc length function for the curve measured from the point P in the direction of increasing t and then reparametrize the curve with respect to arc length starting from P, and (b). find the point 4 units along the curve (in the direction of
> Find the derivative of the vector function. r (t) = sin2at i + tebt j + cos2ct k
> Find a vector equation and parametric equations for the line segment that joins P to Q. P (a, b, c), Q (u, v, w)
> Find a vector equation and parametric equations for the line segment that joins P to Q. P (0, -1, 1), Q ( 1 2 , 1 3 , 1 4 )
> Find a vector equation and parametric equations for the line segment that joins P to Q. P (-1, 2, -2), Q (-3, 5, 1)
> Find a vector equation and parametric equations for the line segment that joins P to Q. P (2, 0, 0), Q (6, 2, -2)
> Find the limit. lim ( te, 1 + t ,t sin 2t3 –
> Find the limit. 1 + t? lim -e-21 1- t2' tant, t
> Find the limit. -i+ /t + 8 j + t sin rt k In t lim
> Find the limit. 12 j+ cos 2t k sin't lim (e-3'i
> Find the derivative of the vector function. r (t) = t sin t i + et cos t j + sin t cos t k
> Find the domain of the vector function. 1 k t - 2 r(t) = cos ti + In tj +
> Find the domain of the vector function. г() — ( In(r + 1), - 2' V9 – t²
> Use the formula in Exercise 63(d) to find the torsion of the curve Exercise 63(d): (r' X r") · r" |r' X r"| (d) т r() = (. 늘?. ).
> Use the Frenet-Serret formulas to prove each of the following. (Primes denote derivatives with respect to t. Start as in the proof of Theorem 10.) (a) r" = s"T + K(s')²N (b) r' X r" = K(s')'B (c) r™ = [s" – k°(s')*]T + [3xs's" + k'(s')*]N + KT(s')°B
> Show that the curvature is related to the tangent and normal vectors by the equation dT KN ds
