need help with practice questions

MCV4U Practice Answer (or Solution) Practice Questions 1. Consider the function f(x) defined by the following graph: 2 1-4 -3 -2 -1 2 Find: a) lim f(x) b) lim f(x) *->-2 x ->- 1 c) lim f (x) d) lim f (x) x -> 1 " 2. Evaluate the following limits. a) lim (x' - 2x2 + 3x -1) b) lim *+ x2 * ->- 1 x-2 x3 - 2 c) lim vsin x d) lim 2 cos.x X->7 /2 3. Consider the function: x+1 if x 1 Find the following limits, if they exit. a) lim f(t) b) lim f (t) x->-1 c) lim f (t) 4. Evaluate. a) lim x - 4 b) lim * +x - 2 *->2 x+2 x- 1 x - 8 c) lim- d) lim x2 - 2x - 3 x-2x2 - 4 x->3x2 - 4x + 3 1 1 e) lim * + 27 f) lim x 2 x->-3 x+3 x-2 x-2 5. Evaluate. a) lim - x -4 b) lim Vx - x x-> 4 \\x - 2 x x-1 V6-x -2 c) lim d) lim V3 +1 - V3 x-2 V3 - x -1 1- 0 e) lim Vx f) lim 1 x x-1 1-0 tv1+t 6. Evaluate. a) lim x -8 b) lim x 3/2 - 8 x-+8 3 x - 2 x-4 x-4 c) lim Vx - 1 x18x-17. Consider the function f(x) defined by the following graph: y 1 - 4 For each value of x = a , classify the function f(x) as continuous, having a jump, a removable or an infinite discontinuity. a) a =-3 b) a =-1 c) a =2 d) a =0 e) a =3 8. For what value of the constant c is the function f (x ) =1 x+c if x - a then find the equation of the tangent line at the given point. a) f (x ) = x2+ x , at P(-1,0) b ) f ( x ) = _ at P(1,0) x+1' 10. For each case find the slope of the tangent line at the general point P(a, f (a)) using m = lim J(ath) - f(a) h - h a ) f (x ) =x2 b ) f (x ) = x3 c) f (x) =x2 - 2x+1 d ) f ( x ) = = 11. For each case, find the ARC over the given interval. a) f ( x ) = x4 - x3 +x 2 , [-1,1] b) f ( x ) = _ 2x - 1 2x + 1 [0,2] 12. For each case, find the IRC at the given number. a) f (x ) = x4 -x', at x = 1 b) f (x ) = - x at x = 0 x 2 + 1 13. For each case, find the average velocity over the given interval. a ) s (t ) = 12 +t , [0,2] b ) s ( t ) = 1 3 -12 , [1,2] 14. For each case, find the instantaneous velocity at the given moment of time a) s(t) = 2t- -t , at t =1 b) s(t) = 213 -3t, at t = 0 Page 2 of 121. Use the rst principles method to nd the derivative of each function. State the domain of each function and its derivative. a) f(x)=3 b) f(x)=72x+5 c) f(x):3x2 2x+1 d) frx):x3 +2):2 +3x+4 e) f(x) = J? 2. For the function y = f(x) defined graphically below, find the values where the function f is no differentiable and the explain why. yA 4 \\ 3 2 I X -4 3 .2 l 0 l 2 3 4' -l 3. Use the power rule to differentiate. a) .x):x h) f(x):x2 c) 10):)? d) f(x)=x e) f 5x x+ 1' J for x E [0,4] c) f(x) = x+-, forxe[1,4] d) f(x) = cosx, for xe[-n/ 2,2x] e) f(x) = xlogx, for x e [1,10] f) f(x) = xe *, for xe [-1,2] g) f(x) = x+ sinx, for xe[0,2x] 7. For each case, find the intervals of concavity. a) f(x) = x4 -6x2 b) f (x) = (x2 - 1) c) f (x ) = - X x 2 - 1 d) f (x ) = (x -1)(x+ 1) 3 e) f(x) = xe f) f (x) = xInx g ) f (x ) = x Inx h) f (x ) = x+ cosx Page 5 of 128. For each case, find the points of inflection. a) frx):x3 x b) f(x) :x+iZ X c) /'(x):(x+1)5\" d) f(x):(1- m1 + 102 e) f(x): x2 lnx f) f(x) : xsin x 9. Find c given that the graph of f(x)=cxZ +l/x2 has a point of inection at (l,f(1)) . 10. Use the second derivative test to nd the local maximum and minimum values of each function. a) f(x):x3 6x2 b) f(x):x4 6x2 5 )5 X c) x): 2 d) f(X) : x +1 (x1)2 11. Find the local minimum and maximum values for: a) y = x3 b) y = x4 6. Second Derivative Eb compute f\"()r) C> find points where f"(x) : O or f"(x) DNE :> nd points of inection :> nd intervals of concavity upward/downward :> check the local extrema using the second derivative test 7. Sketching :> use broken lines to draw the asymptotes c> plot x- and y' intercepts, extrema, and inection points c> draw the curve near the asymptotes c> sketch the curve 12. Sketch the graph of the following polynomial functions. a) f(x) =2):3 73x2 736x b) f(x)=3x5 75x3 c) x): (x 1)3 d) f(X) : x2e + 3) e) f(x) : (x2 3)