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Concept explainers
(a) Find a slope field whose
(b) Prove that if
(c) Find an equation that implicitly defines the integral curve through
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Chapter 8 Solutions
EBK CALCULUS EARLY TRANSCENDENTALS SING
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- Sketch the slope field for y’ + y = 2 at the 25 gridpoints (x, y), where x = 0, 1, ... ,4 and y = 0, 1, ... ,4.arrow_forward2) r(1) = ti -t j-t'k, t20 Draw the graph of the vector-valued function, explaining it in detail.arrow_forwardWhat is the slope of the segment in the slope field for dy/dt= ty + 1 at the point (2, 3)?arrow_forward
- Example: Sketch the direction field for equation: y' = y(2 – y). (1) Evaluate slopes at several y's. y' = y(2 – y) - Y = 3 y' = Y = 2 y' = Y = 1 y' = Y = 0 y' =arrow_forwardFind a vector that gives the direction of maximum rate of increase for the e2y cos x at (n/4,0). (-i+2j) (b) (d) (-i+j) VE (-i-2) (a)({ - ) (c) V2 O a O barrow_forward2. The temperature function on a circular plate R = {(x, y) |r² + y? < 36} is given by T(r, y) = 200 – 2² – y² in degrees Fahrenheit. (a) Sketch (by hand) the gradient field associated with this temperature function. Include level curves for reference, and several vectors. (b) Interpret the gradient field associated with this temperature function. Include ideas about temperature distribution on the plate, the magnitude if the gradient vectors, and changes in temperatures in the plate.arrow_forward
- Let q = - 2xy - y² + 2xz+2yz + z² be a quadratic form on R³ viewed as a polynomial in 3 variables. Find a linear change of variables tou, v, w that puts q into the canonical form in `Sylvester's law of inertia'!. What are the values of the associated indices s, t? Select one: O We let u = √3(x − y + 2), v = x+y, w = (x - y)/2 to find that q = u²+². Hence s = 2, t = 0 in Sylvester's law of intertia. O We let u = x - 2y, v = z+y, w = √3y to find that q=u²v² w². Hence s = 1,t = 2 in Sylvester's law of intertia. O we let u = x, v= √2(x+y), w = x+y+z to find that q = u²v² + w²2. Hence s = 2, t = 1 in Sylvester's law of intertia. O None of the others apply O The quadratic form does not obey the condition to be diagonalisable over R. This is because the minimal polynomial of the corresponding matrix is not a product of distinct linear factors. Hence Sylvester's law of inertia does not apply. By convention, we set s = t = ∞ when this happens.arrow_forward1. (a) Compute the directional derivative of f(x, z) = x²z+xz³ at the point (x, z) = (-2, 4) in the direction û = (3, 1)/√/10. (b) Find the equation of the tangent plane to the surface z = f (x, y) x² - y², at the point (1, 2, -3), in the form ax +by+cz = d. = (c) The function f(x, y, z) = x² + 3xy + 2yz + y² − z² – 11 = 0 defines an implicit function z = z(x, y). Show that the point (x, y, z) = (1, 2, 0) yields f = 0. Find az/ax, dz/dy and evaluate them at the given point (x, y, z) = (1, 2, 0).arrow_forward1) Find the linearization L(x) of f(x) at x = a.f(x) = 2x2 + 4x - 1, a = 4 2) Use implicit differentiation to find dy/dx and d2y/dx2.xy + 3 = y, at the point (4, -1) 3) Provide an appropriate response.The line that is normal to the curve x2 - xy + y2 = 25 at (5, 5) intersects the curve at what other point? 4) Find the second derivative.y = 4x2 + 10x + 2x-3 5) For the given function, find each of the following:(A) Intercepts(B) Vertex(C) Maximum or minimum(D) Rangef(x) = (x + 1)2 - 4 6) Find the slope of the line tangent to the curve y = 3x2 + 9x - 5 at the point (-2, -11). 7) Use implicit differentiation to find dy/dx.cos xy + x4 = y4 Pleas i want answer this Questions (1 -7) Analytical Geometry & Calculusarrow_forward
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