(a) If f (x) = (x2 – 1) / (x2 + 1), find f ‘(x) and f ‘’(x). (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f , f ‘, and f ‘’.
> Find an equation of the plane. The plane through the points (2, 1, 2), (3, -8, 6), and (-2, -3, 1)
> Find an equation of the plane. The plane through the origin and the points (3, -2, 1) and (1, 1, 1)
> Find an equation of the plane. The plane through the points (0, 1, 1), (1, 0, 1), and (1, 1, 0)
> Find an equation of the plane. The plane that contains the line x = 1 + t, y = 2 - t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1
> Find an equation of the plane. The plane through the point (3, -2, 8) and parallel to the plane z = x + y
> Find an equation of the plane. The plane through the point (2, 0, 1) and perpendicular to the line x = 3t, y = 2 - t, z = 3 + 4t
> Find an equation of the plane. The plane through the point (-1, 1 2 , 3) and with normal vector i + 4j + k
> Find an equation of the plane. The plane through the point (5, 3, 5) and with normal vector 2i + j – k.
> Find an equation of the plane. The plane through the origin and perpendicular to the vector 1, −2, 5
> Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. у — 1 y z - 2 L: 1 -1 3 х — 2 L2: у — 3 y -2 7 || 2.
> Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. х — 2 L1: у — 3 z - 1 1 -2 -3 х — 3 у+ 4 z - 2 y 3 L2: 1 -7 ||
> Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. L1: x = 3 + 2t, y = 4 - t, z = 1 + 3t L2: x = 1 + 4s, y = 3 - 2s, z = 4 + 5s
> Find parametric equations for the line segment from (-2, 18, 31) to (11, -4, 48).
> (a) The curve y = x / (1 + x2) is called a serpentine. Find an equation of the tangent line to this curve at the point (3, 0.3). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
> Find a vector equation for the line segment from (6, -1, 9) to (7, 6, 0).
> (a). Find parametric equations for the line through (2, 4, 6) that is perpendicular to the plane x - y + 3z = 7 (b). In what points does this line intersect the coordinate planes?
> Find parametric equations and symmetric equations for the line. The line of intersection of the planes x + -y + 3z = 1 and x - y + z = 1
> Find parametric equations and symmetric equations for the line. The line through (2, 1, 0) and perpendicular to both i + j and j + k
> Find parametric equations and symmetric equations for the line. The line through the points (-8, 1, 4) and (3, -2, 4)
> Two tanks are participating in a battle simulation. Tank A is at point (325, 810, 561) and tank B is positioned at point (765, 675, 599). (a). Find parametric equations for the line of sight between the tanks. (b). If we divide the line of sight into 5
> Let L1 be the line through the points (1, 2, 6) and (2, 4, 8). Let L2 be the line of intersection of the planes P1 and P2, where P1 is the plane x - y + 2z + 1 = 0 and P2 is the plane through the points (3, 2, -1), (0, 0, 1), and (1, 2, 1). Calculate the
> Show that the curve y = 2ex + 3x + 5x3 has no tangent line with slope 2.
> Let L1 be the line through the origin and the point (2, 0, -1). Let L2 be the line through the points (1, -1, 1) and (4, 1, 3). Find the distance between L1 and L2.
> If v1, v2, and v3 are non-coplanar vectors, let (These vectors occur in the study of crystallography. Vectors of the form n1v1 + n2v2 + n3v3, where each ni is an integer, form a lattice for a crystal. Vectors written similarly in terms of k1, k2, and k
> Suppose that a ± 0. (a). If a ∙ b = a ∙ c, does it follow that b = c? (b). If a × b = a × c, does it follow that b = c? (c). If a ∙ b = a ∙ c and a × b = a × c, does it follow that b = c?
> Prove that a.c b. c (а Х Ь) . (с х d) - a·d b.d
> Use Exercise 50 to prove that a × (b × c) + b × (c × a) + c × (a × b) = 0 Exercise 50: Prove Property 6 of cross products, that is, a × (b × c) = (a ∙ c) b – (a ∙ b )c
> Prove Property 6 of cross products, that is, a × (b × c) = (a ∙ c) b – (a ∙ b )c
> Prove that (a – b) × (a + b) = 2 (a × b).
> If a + b + c = 0, show that a × b = b × c = c × a
> For what value of x does the graph f(x) = ex – 2x of have a horizontal tangent?
> (a). Let P be a point not on the plane that passes through the points Q, R, and S. Show that the distance d from P to the plane is (b). Use the formula in part (a) to find the distance from the point P (2, 1, 4) to the plane through the points Q (1, 0,
> Let v = 5j and let u be a vector with length 3 that starts at the origin and rotates in the xy -plane. Find the maximum and minimum values of the length of the vector u × v. In what direction does u × v point?
> (a). A horizontal force of 20 lb is applied to the handle of a gearshift lever as shown. Find the magnitude of the torque about the pivot point P. (b). Find the magnitude of the torque about P if the same force is applied at the elbow Q of the lever.
> A bicycle pedal is pushed by a foot with a 60-N force as shown. The shaft of the pedal is 18 cm long. Find the magnitude of the torque about P. 60 N 70° T10 P
> (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR. P (1, 0, 1), Q (-2, 1, 3), R (4, 2, 5)
> Prove the property of cross products (Theorem 11). Property 4: (a + b) × c = a × c + b × c
> Find a formula for f (n)(x) if f (x) = ln(x - 1).
> The figure shows a vector a in the xy-plane and a vector b in the direction of k. Their lengths are |a | = 3 and |b | = 2. (a). Find |a × b |. (b). Use the right-hand rule to decide whether the components of a × b are positiv
> Find |u × v | and determine whether u × v is directed into the page or out of the page. |v| =5 45° |u| = 4
> Find the work done by a force F = 8 i - 6 j + 9k that moves an object from the point (0, 10, 8) to the point (6, 12, 20) along a straight line. The distance is measured in meters and the force in newtons.
> (a) How is the number e defined? (b) Use a calculator to estimate the values of the limits correct to two decimal places. What can you conclude about the value of e? 2.7h – 1 lim 2.8h – 1 lim and h h
> Reduce the equation to one of the standard forms, classify the surface, and sketch it. x2 + y2 - 2x - 6y - z + 10 = 0
> Small birds like finches alternate between flapping their wings and keeping them folded while gliding (see Figure 1). In this project we analyze this phenomenon and try to determine how frequently a bird should flap its wings. Some of the principles are
> If g(x) = x/ex, find g(n)(x).
> Use logarithmic differentiation to find the derivative of the function. y = x cos x
> If f (x) = x2 / (1 + x), find f “(1).
> Use logarithmic differentiation to find the derivative of the function. y = xx
> (a) If f (x) = (x2 – 1) ex, find f ‘(x) and f ‘’(x). (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f , f ‘, and f ‘’.
> Differentiate. y = 1 t 3 + 2 t 2 – 1
> Differentiate the function. B(y) = cy-6
> Differentiate. y = x2 + 1 / x3 + 1
> Differentiate. f (z) = (1 – ez)(z + ez)
> Find dy/dx by implicit differentiation. 2x2 + xy - y2 = 2
> Differentiate. J(v) = (v3 - 2v)(v-4 + v-2)
> Differentiate the function. g(x) = x2 (1 -2x)
> Sketch the parabolas y = x2 and y = x2 - 2x + 2. Do you think there is a line that is tangent to both curves? If so, find its equation. If not, why not?
> If c > ½, how may points through the point (0,c) are normal lines to the parabola y = x2? What if c ≤ ½ ?
> Draw a diagram showing two perpendicular lines that intersect on the y-axis and are both tangent to the parabola y = x2. Where do these lines intersect?
> A tangent line is drawn to the hyperbola xy = c at a point P. (a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is P. (b) Show that the triangle formed by the tangent line and the coordinate axes always has
> Let Find the values of m and b that make f differentiable everywhere. -2 if x<2 f(x) = тx + b ifx>2
> The graph of any quadratic function f(x) = ax2 + bx + c is a parabola. Prove that the average of the slopes of the tangent lines to the parabola at the endpoints of any interval [ p, q ] equals the slope of the tangent line at the midpoint of the interva
> (a) If f (x) = ex / (2x2 + x + 1), find f ‘(x). (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f ‘.
> What is the value of c such that the line y − 2x + 3 is tangent to the parabola y = cx2?
> Use logarithmic differentiation to find the derivative of the function. y = (x2 + 2)2 (x4 + 4)4
> Differentiate the function. f(t) = 1.4t5 – 2.5t2 + 6.7
> For what values of a and b is the line 2x + y = b tangent to the parabola y = ax2 when x = 2?
> Suppose the y = x4 + ax3 + bx2 + cx + d curve has a tangent line when x = 0 with equation y = 2x + 1 and a tangent line when x − 1 with equation y = 2 - 3x. Find the values of a, b, c, and d.
> Find the parabola with y = ax2 + bx equation whose tangent line at (1, 1) has equation y = 3x - 2.
> At what numbers is the following function g differentiable? Give a formula for g’ and sketch the graphs of g and g’. 2х if x<0 9(х) — { 2х — х? if 0 <x<2 2 — х if x> 2
> Let Is f differentiable at 1? Sketch the graphs of f and f ’. (x² + 1 if x <1 f(x) x + 1 if x>1
> Find a parabola with y = ax2 + bx + c equation that has slope 4 at x = 1, slope -8 at x = -1, and passes through the point (2, 15).
> Find a cubic function y = ax3 + bx2 + cx + d whose graph has horizontal tangents at the points (-2, 6) and (2, 0).
> The equation y’’ + y’ – 2y = x2 is called a differential equation because it involves an unknown function y and its derivatives y’ and y”. Find constants A, B, and C such that the function satisfies y = Ax2 + Bx + C this equation. (Differential equations
> (a) If f (x) = (x3 – x)ex, find f ‘(x). (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f ‘.
> Let f (x) = logα (3x2 - 2). For what value of b is f ‘(1) = 3?
> Differentiate the function. f(t) = 2t3 – 3t2 – 4t
> Use the Chain Rule and the Product Rule to give an alternative proof of the Quotient Rule.
> Find a second-degree polynomial P such that P(2) = 5, P’(2) = 3, and P’’(2)= 2.
> Use the Chain Rule to prove the following. (a) The derivative of an even function is an odd function. (b) The derivative of an odd function is an even function.
> Computer algebra systems have commands that differentiate functions, but the form of the answer may not be convenient and so further commands may be necessary to simplify the answer. (a) Use a CAS to find the derivative in Example 5 and compare with the
> The table gives the US population from 1790 to 1860. (a) Use a graphing calculator or computer to fit an exponential function to the data. Graph the data points and the exponential model. How good is the fit? (b) Estimate the rates of population growth
> The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The following data describe the charge Q remaining on the capacitor (measured in microcoulombs, mC) at time t (measured in seconds).
> Air is being pumped into a spherical weather balloon. At any time t, the volume of the balloon is V(t) and its radius is r(t). (a) What do the derivatives dV/dr and dV/dt represent? (b) Express dV/dt in terms of dr/dt.
> Differentiate the function. R(a) = (3a + 1)2
> How many tangent lines to the curve y = x/(x + 1) pass through the point (1, 2)? At which points do these tangent lines touch the curve?
> In Section 1.4 we modeled the world population from 1900 to 2010 with the exponential function P(t) = (1436.53) . (1.01395)t where t = 0 corresponds to the year 1900 and Pstd is measured in millions. According to this model, what was the rate of increase
> The average blood alcohol concentration (BAC) of eight male subjects was measured after consumption of 15 mL of ethanol (corresponding to one alcoholic drink). The resulting data were modeled by the concentration function C(t) = 0.0225te-0.0467t where t
> Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs. Ка) f(x) — х" (b) f(x) = 1/x
> The motion of a spring that is subject to a frictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine function. Suppose the equation of motion of a point on suc
> If the equation of motion of a particle is given by s = A cos (ωt + δ), the particle is said to undergo simple harmonic motion. (a) Find the velocity of the particle at time t. (b) When is the velocity 0?
> The displacement of a particle on a vibrating string is given by the equation s(t) = 10 + 1/4 sin(10πt) where s is measured in centimeters and t in seconds. Find the velocity of the particle after t seconds.
> Let f (x) = cx + ln (cos x). For what value of c is f ‘( π/4) = 6?
> Find the 1000th derivative of f (x)=xe-x.
> For what values of r does the function y=erx satisfy the differential equation y’’ - 4y’ + y=0?
> Show that the function y=e2x (A cos 3x + B sin 3x) satisfies the differential equation y’’ - 4y’ + 13y=0.
> If F(x) = f (x f (x f (x))), where f (1) = 2, f (2) = 3, f ‘(1) = 4, f ‘(2) = 5, and f ‘(3) = 6, find f’(1).
> If F(x) = f (3 f(4 f (x))), where f (0) = 0 and f ‘(0) = 2, find F’(0).
> If g is a twice differentiable function and f (x) = xg(x2), find f ‘’ in terms of g, g’, and g’’.
> Let r(x) = f (g(h(x))) , where h(1) = 2, g(2) = 3, h’(1) = 4, g’(2) = 5, and f ‘(3) = 6. Find r’(1).
