Math 2413 Exam 2 Litong Name Test No

Transcription

1 Math 413 Eam Litong Name Test No Find the equation for the tangent to the curve at the given point. 1) f() = - ; (1, 0) C) The graph of a function is given. Choose the answer that represents the graph of its derivative. ) D) A) Given the graph of f, find an values of at which f is not defined. 3) B)

2 The figure shows the graph of a function. At the given value of, does the function appear to be differentiable, continuous but not differentiable, or neither continuous nor differentiable? 4) = The function s = f(t) gives the position of a bod moving on a coordinate line, with s in meters and t in seconds. 11) s = 6t + 4t + 3, 0 t Find the bodʹs speed and acceleration at the end of the time interval. The graphs show the position s, velocit v = ds/dt, and acceleration a = ds/dt of a bod moving along a coordinate line as functions of time t. Which graph is which? 1) - -4 t Find the derivative. BAC ) w = z-6-1 z Find the second derivative. 6) = Find. 7) = ( )(3 - + ) Find the derivative of the function. 8) r = θ - θ + Find the derivative. 13) s = t6 tan t - t 14) r = 16 - θ8 cos θ Find the indicated derivative. ) Find (4) if = 4 sin. 16) Find the tangent to = cos at = π. Suppose u and v are differentiable functions of. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 9) u(1) =, u (1) = -6, v(1) = 7, v (1) = -. d d u v at = 1 ) Find an equation for the tangent to the curve 18 = at the point (1, 6). + 17) Find all points on the curve = sin, 0 π, where the tangent line is parallel to the line = 1. 18) Find all points on the curve = cos, 0 π, where the tangent line is parallel to the line = -. Find the derivative of the function. 19) = cos6(πt - 14)

3 Find. 0) h(θ) = + sin(1θ) 1) = cot 6 Suppose that the functions f and g and their derivatives with respect to have the following values at the given values of. Find the derivative with respect to of the given combination at the given value of. f() g() f () g () ) ) One airplane is approaching an airport from the north at 17 km/hr. A second airplane approaches from the east at 8 km/hr. Find the rate at which the distance between the planes changes when the southbound plane is 3 km awa from the airport and the westbound plane is 16 km from the airport. Round to the nearest kilometer per hour. g(), = 3 Use implicit differentiation to find d/d. 3) = 8 4) + 1 = 4 ) cos + 7 = 7 6) = sec(7) At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 7) = + 11, tangent at (0, 1) 8) The area of the base B and the height h of a pramid are related to the pramidʹs volume V b the formula V = 1 Bh. How is dv/dt 3 related to dh/dt if B is constant? Round our answer, if appropriate. 9) As the zoom lens in a camera moves in and out, the size of the rectangular image changes. Assume that the current image is cm 3 cm (length width ). Find the rate at which the area of the image is changing (da/df) if the length of the image is changing at 0. cm/s and the width of the image is changing at 0. cm/s. 3

4 Answer Ke Testname: MA413XREVIEW 1) = 1-1 ) B 3) = 4) Neither continuous nor differentiable ) -6z z 6) ) ) r = θ( θ + ) 9) ) = ) 8 m/sec, 1 m/sec 1) C = position, B = velocit, A = acceleration 13) ds dt = t 6 1 sec t + 6t tan t - t 14) dr dθ = - 8θ 7 cos θ + θ8 sin θ ) (4) = 4 sin 16) = - + π 17) 18) π 3, 3, π 3, - 3 π, 0 19) d dt = -6π cos (πt - 14) sin(πt - 14) 0) h (θ) = 1) 6 cos(1θ) + sin(1θ) 18 csc 6 cot 6 ) 7 4 3) ) - ( + 1) ) sin 76 + sin 6) 1 7 cos(7) cot(7) 4

5 Answer Ke Testname: MA413XREVIEW 7) = ) dv dt = B dh 3 dt 9). cm/sec 30) -80 km/hr