Continuity of a Function [-1 if x0 not. (Answer: f is not continuous at 0) 2. Let f(x)= x if x*0,F 1 if x=0. " For each real number a, determine whether f is continuous or discontinuous at a. (Answer: f is not continuous at 0, and f is continuous at a) The Derivative Find f'(x) of the following equations f (x) = - 8 3. x- 2' f' (xx ) = - Ni Answer : 4. f(x) = sin x;x,=0 Answer : f'(x) =1 5. f(x) = sin x;x=n Answer: f'(x) =0 6. g(t) = - Answer : g'(t) =- 1+1 (1 + 1)2 5 7. R(z) = V5z-8 Answer : R' (z) =- 2V/5z-8 The Slope 8. Find the derivative of equation f(x) =x2 +4x-1 (Answer: f'(x) =2x+4) 9. Find an equation of the tangent line at the given point. a) y= 9-x2; (2,5) Answer : y = -4x +13 b) y = 2x2 + 4x; (-2,0) Answer : y = -4x-8 c) y=x' + 3; (1,4) Answer : y = 3x+1 10. Find the slope of an equation f(x) =3x2 -12x+8 (Answer: m(x, ) = 6.x, -12)Rate of Chggge 11. Let V(x) cubic centimeters be the volume of a cube having an edge of x centimeters, measured to four significant digits. On a calculator compute the average rate of change of V(X) with respect to x as x changes from (a) 3.000 to 3.200; (b) 3.000 to 3.100: (c) 3.000 to 3.010: (d) 3.000 to 3.001. (e) What is instantaneous rate of change of V(X) with respect to x when x is 3.000? (Answer: (a) 28.84: (b) 27.91: (c) 27.09; (d) 27.01; (e) 27 12. Let m(x) be the slope of the tangent line to the curve }H=x32x24x at the point (x.y). Find the instantaneous rate of change of m(x) with respect to x at the point (2,2). (Answer: 8) 13. Let A(x) square centimeters be the area of a square having a side of x centimeters, measured to four significant digits. Calculate the average rate of change of A(x) with respect to x as X changes from (a) 4.000 to 4.600: (b) 4.000 to 4.300: (C) 4.000 to 4.100; (d) 4.000 to 4.050. (e) What is the instantaneous rate of change of A(x} with respect to x when x is 4.000? (Answer: (a) 8.6; (b) 8.3; (c) 8.1: (d) 8.05; (e) 8 The Chain Rule 14. Compute 2x+ 1 dx (Answer : 20(2x + 1)3 3x - 1 (3x -1)5 15. Find F' (t) if F(t) = tan(3t2 + 2t) (Answer: 2(3t + 1) sec-(312 + 2t) ) dy 16. Find dx if y = sin(cos x) (Answer: - sin x[cos(cos x)]) 17. Compute D. (sec4 2x2) (Answer: 16xsec4 2x2 tan 2x2) 18. Solve f(x) = sin(3x2 +x) (Answer: f'(x) = (6x+1)cos(3x2 +x) ) The General Power Rule 19. Find (Answer : - 15 20. Suppose f(x) =15x* . Find f' (x) (Answer: 60x3 ) 21. Suppose f(x) =2x' + 16x2 -5x+4. Find f' (x). (Answer: 6x2 + - x-5) 22. Suppose f(x) = x +4x -3x3. Find f' (x). (Answer: x3 +4x 6 -3x3) 23. Suppose f(x) = Vx+-. Find f' (x). Vx (Answer : VX VX 4x x 2