Sample of Material from MAT181 at Broome Community College This material is for sample purposes only and is not to be considered as an official listing of topics. 1. Set up a table of values to evaluate )1xsin(1xlim31x−−→ accurate to 8 decimal places. 2. Evaluate the following limits algebraically or with L'Hopital's Rule a) xsine1limx20x−→ b) 24747limxxx+−∞→ 3. Using your answer from 2), explain whether 2x474x7)x(f+−=has a horizontal asymptote. If so, state the equation. 4. Use the limit definition of the derivative to help find the equation of the tangent line to f(x) = x + x1 when x = 10. 5. Using the Power, Product and Quotient rules find the derivative for the following functions. Factor and/or reduce where possible. a) f(x) = sin2 x b) y = xlnx c) h(x) = d) p(x) = ln (secx) xtane2x 6. Graph a single function that has all of the following properties. a) f(0) = -1; f(-5) = 0 b) and −∞=→−)x(f2xlim∞=→+)x(f2xlim c) f ' (1) = 0 and f ' (4) = 0 d) ; ∞=∞→)x(fxlim∞=−∞→)x(fxlim e) f ' (x) < 0 on (-∞ , 1) and (1 , 2) and (2 , 4) ; f ' (x) > 0 on (4 , ∞). f) f '' (x) < 0 on (1 , 2) ; f '' (x) > 0 on (-∞ , 1) and (2 , ∞). 7. Find the equation of the tangent line to (sinx) y2 + 5y = x + 10 at the point (0, 2).Net 8. Use derivatives to find the critical values for f(x) = 5x8x4124++, then set up a table to find the extreme values on [-2, 5] . 9. A fishing boat lays a circular net in the water. If the netting is pulled in at 20 ft/minute, how fast is the radius of the circle decreasing when the diameter of the net is 100 feet? Set up an equation and show a Calculus answer. 10. Evaluate the following antiderivatives. Show the Change of Variables in each case. a) b) ∫+ dx)4x(secx322∫−dxxsin2xcos 11. Find the area of the region bounded by f(x) = 8 – x2 and y = 2x . Evaluate the integral using the Fundamental Theorem of Calculus. 12. Find the volume of the solid formed when the region trapped between y = x2 , y = 0 and x = 3 is rotated around the line x = 3. State the method you are using. 13. A car traveling 60 feet per second accelerates at 20 feet per second2. a) Use calculus to derive a position function. Note: you must derive the function, not copy a finished one. b) How long until the car travels 5280 feet? Show