NAME____________________________________ SCHOOL__________________________________ TEST 1 Monday, September 21, 2009 Fax your answers to: Irvin Roy Hentzel at Fax 515-294-5454 Snail Mail your answers to: Irvin Roy Hentzel Department of Mathematics 432 Carver Hall Iowa State University Ames, Iowa 50011-2064 1. The speed of two drag racers consists of joined line segments. The vertices of the graphs are. racer A (0,0) (2,40) (4,65) (6,80) racer B (0,0) (2,45) (4,55) (6,80) (a) How far did racer A go in 6 seconds? 40 + 105 + 145 = 290 (b) How far did racer B go in 6 seconds? 45 + 100 + 135 = 280 (c) Which racer was ahead after six seconds? A was ahead (d) At what times did they pass? at 4 seconds both had gone 145 feet 2. Find the following limit. Show all of your work. _ _ _ _ | 2 | | | Sqrt| x - 4 | - Sqrt| x+2 | |_ _| |_ _| Limit ---------------------------- x->3 2 x - 9 _ _ _ _ | 2 | | | | x - 4 | - | x+2 | |_ _| |_ _| Limit ------------------------------------------------- x->3 / _ _ _ _ \ / | 2 | | | \ ( Sqrt| x - 4 | + Sqrt| x+2 | )(x+3)(x-3) \ |_ _| |_ _| / \ / 2 x - x - 6 Limit ------------------------------------------------- x->3 / _ _ _ _ \ / | 2 | | | \ ( Sqrt| x - 4 | + Sqrt| x+2 | )(x+3)(x-3) \ |_ _| |_ _| / \ / (x-3)(x+2) Limit --------------------------------------------- x->3 / _ _ _ _ \ / | 2 | | | \ ( Sqrt| x - 4 | + Sqrt| x+2 | )(x+3)(x-3) \ |_ _| |_ _| / \ / (x+2) Limit --------------------------------------------- x->3 / _ _ _ _ \ / | 2 | | | \ ( Sqrt| x - 4 | + Sqrt| x+2 | )(x+3) \ |_ _| |_ _| / \ / 5 ------------ 2 Sqrt[5] 6 Sqrt[5] --------------- 12 f[x_] := (Sqrt[x^2-4]-Sqrt[x+2])/(x^2-9); Limit[f[x],{x->3}]; p1 = Plot[f[x],{x,2,4}]; p2 = Plot[Sqrt[5]/12,{x,2,4}]; Show[p1,p2] 3. Find the equation of the tangent line to 4 3 2 y = x + x + x + 2 at the point (2,30). y' = 4x^3 + 3x^2 + 2x y'(2) = 32 + 12 + 4 = 48 y-30 = 48(x-2) f[x_] := x^4 + x^3 + x^2 + 2; p1 = Plot[f[x],{x,-2,3}]; p2 = Plot[30+48(x-2),{x,-2,3}]; Show[p1,p2] 4. Find the points (x,y) where the tangent line 3 2 to y = x + x - x + 1 is horizontal. y' = 3x^2 + 2 x - 1 = (3x-1)(x+1) x= 1/3 and x = -1 (1/3,22/27) and (-1,2) f[x_] := x^3 + x^2 - x + 1 p1 = Plot[f[x],{x,-2,2}]; p2 = ListPlot[{{1/3,22/27},{-1,2}}]; Show[p1,p2] 5. Explain why calculus uses radians instead of degrees. Sin[x] The limit -------- = 0 when x is in radians. x->0 x If calculus were done using degrees, this limit would be a constant different than one and would complicate matters. 6. Give the definition of the derivative of a function y = f(x). / f(x+h)-f(x) f (x) = Limit -------------- h->0 h 7. A spherical balloon is filling at the rate of Pi cubic feet per minute. At what rate is the radius changing when the balloon holds 36 Pi Cubic feet. V = 4/3 Pi r^3 When V = 36 Pi, then 36 Pi = 4/3 Pi r^3 108 = 4 r^3 27 = r^3 3 = r dV/dt = 4 Pi r^2 dr/dt When V = 36 Pi and r = 3 pi = 4 Pi 9 dr/dt dr/dt = 1/36 ft/min. <========== Answer 8. (a) State the chain rule. / / / (fog)(x) = f (g(x) ) g (x) (b) Prove the chain rule. f(g(x+h)) - f(g(x)) Limit --------------------- h->0 h f(g(x+h)) - f(g(x)) g(x+h)-g(x) Limit --------------------- * ---------------- h->0 g(x+h) - g(x) h / / f (g(x)) g (x) In problems 9 through 20 Find the derivatives. Do NOT simplify 3 9. y = 9 x + 2 x + 1 / 2 y = 27 x + 2 2 2 x + x 10. y = -------------- 3 x - 1 3 2 2 / (x - 1)(4x+1) - (2 x + x) 3 x y = ------------------------------------ 3 2 (x - 1 ) 5 11. y = x Tan[x] / 4 5 2 y = 5 x Tan[x] + x Sec [x] 12. y = Tan[ Tan[x] ] / 2 2 y = Sec [ Tan[x] ] Sec [x] 4 13. y = ( Sin[x] ) / 3 y = 4 ( Sin[x] ) Cos[x] 5 14. y = ( Sin[5 x] ) / 4 y = 5 (Sin[5 x]) Cos[5x] 5 2 4 1/2 15. y = ((x + 5) + 7) / 2 4 -1/2 2 3 y = 1/2 ((x + 5) + 7) 4( x + 5 ) 2x Sin[ 2x ] 16. y = -------- Sin[ x ] / Sin[x] Cos[2x]2 - Sin[2] Cos[x] y = ------------------------------ 2 Sin [x] Tan[Pi-1] 17. y = ------------- Pi -1 / y = 0 3 5/2 18. y = (x + 1) / 3 3/2 2 y = 5/2 (x + 1) 3 x 3 3 19. y = ( 2 x + 7 x - 1 ) / 3 2 2 y = 3 (2 x + 7 x - 1 ) ( 6 x + 7 ) 3 4 20. y = 3 ( Sec[x ] ) / 3 3 3 3 2 y = 12 ( Sec[x ] ) Sec[x ] Tan[x ] 3 x