Find the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x

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1 Assignment 5 Name Find the indicated derivative. ) Find y(4) if y = sin x. ) A) y(4) = cos x B) y(4) = sin x y(4) = - cos x y(4) = - sin x ) y = (csc x + cot x)(csc x - cot x) ) A) y = 0 B) y = y = - csc x y = - csc x cot x ) s = sin t - e-t ) A) ds dt = -cos t + e -t ds dt = cos t + e -t B) ds dt = -cost - e -t ds dt = cost - e -t Solve the problem. 4) Does the graph of the function y = x + 4 sin x have any horizontal tangents in the interval 4) 0 x? If so, where? A) Yes, at x =, x = B) Yes, at x = No Yes, at x =, x = 4 5) s = t - cos t + 4et 5) A) ds dt = t - sin t + 4e t ds dt = t + sin t + 4e t B) ds dt = t + sin t + 4e t ds dt = t + sin t - 4e t The equation gives the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). ) s = 8 sin t - cos t ) Find the body's acceleration at time t = /4 sec. A) - 9 m/sec B) - 7 m/sec 9 m/sec 7 m/sec

2 7) p = 9 + sec q 9 - sec q 7) A) dp dq = 8 sin q (9 cos q - ) dp dq = 8 tan q (9 - sec q) B) dp dq = - 8 sin q (9 cos q - ) dp dq = - sec q tan q (9 - sec q) 8) Graph y = - tan x and its derivative together on -, y = - tan x ever positive? Explain.. Is the slope of the graph of 8) Find the indicated derivative. 9) Find y if y = - cos x. 9) A) y = sin x B) y = - cos x y = - sin x y = cos x Find the limit. 0) lim sin x / + cot x cos x + sin x 0) A) - B) 0 4 Find y. ) y = tan(8x - ) ) A) 8 sec (8x - ) B) 8 sec (8x - ) tan(8x - ) sec(8x - ) sec (8x - ) tan(8x - ) Find an equation for the line tangent to the curve at the point defined by the given value of t. ) x = sin t, y = sin t, t = ) A) y = x + B) y = -x + y = x y = x - Find the value of dy/dx at the point defined by the given value of t. ) x = 9t -, y = t, t = ) A) B)

3 Find y. 4) y = x 4 4) A) 9 x x - x x + x x B) - 48 x x x + x x Find the value of (f g) at the given value of x. 5) f(u) = u -, u = g(x) = x, x = 4 5) u + A) 8 B) Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x f(x) g(x) f (x) g (x) ) ) g(x + f(x)), x = A) -4 B) Given y = f(u) and u = g(x), find dy/dx = f (g(x))g (x). 7) y = 5, u = 4x - 7) u A) - 40 (4x - ) B) - 0 4x - 40x 4x x - Find the value of dy/dx at the point defined by the given value of t. 8) x = sin t, y = cos t, t = 4 8) A) B) - - 9) Suppose that u = g(x) is differentiable at x = and that y = f(u) is differentiable at u = g(). 9) If the tangent to the graph of y = f(g(x)) at x = is not horizontal, what can we conclude about the tangent to the graph of g at x = and the tangent to the graph of f at u = g()? Explain.

4 Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x f(x) g(x) f (x) g (x) 0) 8 0) /g(x), x = 4 A) 4 9 B) Solve the problem. ) Find the points on the curve x - xy + y = where the tangent is parallel to the y-axis. ) A) (-, -4), (, 4) B) (-, -4), (-, ), (, -), (, 4) (-4, -), (-, ), (, -), (4, ) (-4, -), (4, ) ) xy = 4, slope at (, ) ) A) - 4 B) - - ) 5xy - cos y =, slope at (, ) ) A) - B) 0 - Assuming that the equations define x and y implicitly as differentiable functions x = f(t), y = g(t), find the slope of the curve x = f(t), y = g(t) at the given value of t. 4) x cos t + x = t, y = t sin t + t, t = 4) A) + B) + + 5) y4 + x = y + 9x, tangent at (0, ) 5) A) y = - x B) y = x - y = 9 x + y = 9 4 x + Use implicit differentiation to find dy/dx. ) ex = sin(x + 4y) ) A) dy dx = - ex 4sin(x + 4y) dy dx = 8ex sin(x + 4y) B) dy dx = ex 4sin(x + 4y) dy dx = -ex sin(x + 4y) 4

5 7) Given xy + yx = 0, find both dy/dx (treating y as a differentiable function of x) and 7) dx/dy (treating x as a differentiable function of y). How are dy/dx and dx/dy related? Assuming that the equations define x and y implicitly as differentiable functions x = f(t), y = g(t), find the slope of the curve x = f(t), y = g(t) at the given value of t. 8) tx + 4t = 4, y - t - t = 0, t = 8) A) B) Use implicit differentiation to find dy/dx. 9) y cos y = x + y 9) A) - y sin y cos y - B) y sin y + y cos y - y y sin y - y sin y + y cos y - 0) x5y5 =, normal at (, ) 0) A) y = - x + B) y = x y = -x + 5 y = x - 5

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