FINAL EXAM Math 2413 Calculus I

Transcription

1 FINAL EXAM Math Calculus I Name ate ********************************************************************************************************************************************** MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit. lim A B oes not eist C Find the limit, if it eists. lim A 0 B C lim - cos A B - C 0 cos lim - + A 0 B - C oes not eist lim e - A B C 0 A-

2 Find the equation for the tangent to the curve at the given point. f( = + ; (-, A = 9 + B = -9-0 C = = - f( = +, (, A = + B = - C = + = +9 Solve the problem. The velocit of water ft/s at the point of discharge is given b v =.9 P, where P is the pressure lb/in of the water at the point of discharge. Find the rate of change of the velocit with respect to pressure if the pressure is 0.00 lb/in. A 9. ft/s per lb/in B 0.0 ft/s per lb/in C. ft/s per lb/in 0. ft/s per lb/in Find the slope of the tangent line at the given value of the independent variable. 9 g( = +, = A B C Find the indicated derivative. 0 Find ( if = - cos. A ( = - sin B ( = cos C ( = - cos ( = sin 0 Solve the problem. Find the tangent to = cos at = π. A = - + π B = + π C = = - - π Find the tangent to = - sin at = π. A = - + π - B = - C = - π + = - + A-

3 Find the limit. lim cos - A B 0 C - Given = f(u and u = g(, find d/d = f (g(g (. = u, u = - A - B - - C ( - = u(u -, u = + A + - B + + C Write the function in the form = f(u and u = g(. Then find d/d as a function of. = A = u0; u = - B = u0; u = - C = u0; u = - - ; d d = ; d d = ; d d = = u - u - u; u = 0; d d = = csc(cot A = csc u; u = cot ; d = - csc(cot cot(cot d B = cot u; u = csc ; d d = csc (csc csc cot C = csc u; u = cot ; d d = csc cot = csc u; u = cot ; d d = csc(cot cot(cot csc A-

4 Solve the problem. The position of a particle moving along a coordinate line is s = + t, with s in meters and t in seconds. Find the particle's velocit at t = sec. m/sec A m/sec B m/sec C - m/sec 9 Suppose that the velocit of a falling bod is v = ks (k a constant at the instant the bod has fallen s meters from its starting point. Find the bod's acceleration as a function of s. A a = ks B a = ks C a = ks a = ks 9 Use implicit differentiation to find d/d. 0 - = A B - - C = + A ( ( - B ( ( - C ( ( - ( ( - Use implicit differentiation to find d/d and d/d. - + = A d d = - + ; d d = - ( + C d d = ; d d = + ( + B d d = + + ; d d = + ( + d d = ; d d = - ( + 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. + = + 0, slope at (0, A 0 B 0 C - A-

5 Assuming that the equations define and implicitl as differentiable functions = f(t, = g(t, find the slope of the curve = f(t, = g(t at the given value of t. + / = t + t, (t + - t =, t = 0 A - B - C - Solve the problem. Water is falling on a surface, wetting a circular area that is epanding at a rate of mm/s. How fast is the radius of the wetted area epanding when the radius is mm? (Round our answer to four decimal places. A 0.0 mm/s B 0.00 mm/s C.0 mm/s 0.0 mm/s A wheel with radius m rolls at rad/s. How fast is a point on the rim of the wheel rising when the point is π/ radians above the horizontal (and rising? (Round our answer to one decimal place. A.0 m/s B.0 m/s C.0 m/s.0 m/s You want a linearization that will replace the function over an interval that includes the point 0. To make our subsequent work as simple as possible, ou want to center the linearization not at 0 but at a nearb integer = a at which the function and its derivative are eas to evaluate. What linearization do ou use? f( =, 0 =.9 + A + B + 0 C - 0 f( = +, 0 =. A + B + C + + Use Newton's method to estimate the requested solution of the equation. Start with given value of 0 and then give as the estimated solution = 0; 0 = -; Find the one real solution. A -0. B -0. C A-

6 Find the value of the specified finite sum. n n n 0 Given ak = - and bk =, find ak - bk. k= k= k= A B C Graph the function f( over the given interval. Partition the interval into subintervals of equal length. Then add to our sketch the rectangles associated with the Riemann sum f(ck Δk, using the indicated point in the kth k= subinterval for ck. f( = +, [0, ], left-hand endpoint A 0.. B C A-

7 Find the formula and limit as requested. For the function f( = -, find a formula for the lower sum obtained b dividing the interval [0, ] into n equal subintervals. Then take the limit as n to calculate the area under the curve over [0, ]. A - 0n + n + n ; Area = n C + 0n + n + ; Area = n B 0n + n + n ; Area = n - 0n + n + n ; Area = n Solve the problem. Suppose that A -; - f( d = -. Find 9f(u du and - f(u du. B 9; C ; - -; Suppose that - g(t dt = -. Find g( - d - and - g(t dt. - A 0; - B -; C ; - ; Graph the integrand and use areas to evaluate the integral. (- + 0 d -0 A B 0 C 00 ( + d - A B 0 C Evaluate the integral d A - B C - A-

8 Find the average value of the function over the given interval. f( = on [-0, ] A 0 B C Use the substitution formula to evaluate the integral d 0 A B C π cos sin d π/ A B - C - 0 A-

9 Answer Ke Testname: UNTITLE C A C B A 9 C 0 C A C C A B B 9 C 0 A A B A B A B C 9 0 A A C A C 9 B 0 B A-9