(d) lim [p(x)]f(x) = 1. (1 point) x-+al Suppose that lim f(x) = 0, limg(x) = 0, limh(x) = x-+a 1, lim p(x) = 0o, and lim q(x) = co. (e) lim [p(x) 19(x) = Evaluate each of the following limits. (a) lim If (x)p(x)] = (f) lim p(x) = Note: Input DNE, infinity, and -infinity for does not exist , (b) lim [h(x)p(x)] = co, and -co, respectively. If the result is indeterminate, enter ibili/bi. 3. (1 point) Evaluate the following limit. lim 21nx (c) lim [p(x)q(x)] = Note: Input DNE, infinity, and -infinity for does not exist , co, and -co, respectively. Note: Input DNE, infinity, and -infinity for does not exist , 4. (1 point) Evaluate the following limit. co, and -co, respectively. If the result is indeterminate, enter bili/bi. lim sin ( * ) 2. (1 point) Suppose that lim f(x) = 0, limg(x) = 0, limh(x) = Note: Input DNE, infinity, and -infinity for does not exist , x-+a 1, lim p(x) = co, and lim q(x) =09. po, and -co, respectively. x -+a 5. (1 point) Evaluate the following limit. Evaluate each of the following limits. lim xsex2 (a) lim [f (x)]8(x) = Note: Input DNE, infinity, and -infinity for does not exist , x-+al co, and -oo, respectively. 6. (1 point) Evaluate the following limit. (b) lim If (x)]P(x) = lim x-too (1+ 2 ) x-ta Note: Input DNE, infinity, and -infinity for does not exist , co, and -co, respectively. 7. (1 point) Evaluate the following limit. (c) lim [h(x) ]P(x) = lim x8 sin(x) = Note: Input DNE, infinity, and -infinity for does not exist , co, and -co, respectively.8. (1 point) For what values of a and I) does lim ( sin(2.x) b) _ x>D 0? x3 +a+x_2 b: 9. (1 point) Consider the function f(x} = x2 4x+ 9 on the interval [0,4]. (a) What conditions must hold true in order to apply Rolle's Theorem? f(x) is '? on [0,4]; f(x} is '? on (0,4); and u) = N) = . (b) Since part (a) holds true for f (x), by Rolle's theorem, there exists a e such that: NC) =- (c) Find c. C: 10. (1 point) Use Rolle's Theorem and a proof by contra- diction to show that the function f (x) = 8x7 +4x 18 does not have two real roots. ibLProofii'bgl: Suppose f (x) has two real roots a and b such that f (a) = f (b) = Since the conditions of Rolle's theorem hold true for f on [a,b], there exists at least one number c in the interval ((1,3)) such that f'(c) = . However, the derivative f'(x) 2 therefore, it is for j"1 (x) 2. This contradicts the conclusion of Rolle's Theorem and, there- fore, f have two real roots. is always and, 11. (1 point) Consider the function f (x) = 5x3 4x on the interval [4,4]. (a) The slope of the secant line joining (4,f(4}} and (4,114)) is m : (b) Since the conditions of the Mean Value Theorem hold true, there exists at least one c on (4,4) such that f'(C) =- (c) Find c. c : Note: If there is more than one answer, separate them with acomma. 12. (1 point) Find all critical numbers 6 of f0) = 4:2{3 +t5fi3. 6: Note: If there is more than one critical number, separate them by a comma. 13. (1 point) Suppose f (x) = Ate9", 0 S x g 2. Use the Closed Interval Method to nd points at which the absolute extrema of f occur. ) } Absolute minimum: ( Absolute maximum: ( 6x2 14. (1 point) Consider the function f (x) = x 4. (a) f is increasing for x E (b) f is decreasing forx E (c) The local maxima of f occur at x = (d) The local minima of f occur at x = Note: Input U, innity, and -innity for union, 00, and oo, respectively. If there are multiple answers, separate them by commas. If there is no answer, input none. 1 15. (1 point) Consider the function f (x) =3 3x2 + 4' (a) f"(x) > on I. (a) f is concave up for x E (b) 8"(x) = 2(A2 + Bf"(x)), where A = - and B = (c) 8" (x ) > . on I. (b) f is concave down for x E (d) g(x) is on I. (c) The inflection points of f occur at x = Note: Input CU, ibjCD;/bi, ibif(x)i/bi, ibif'(x)i/bi, and ibif"(x);/bi for concave up, concave down, f(x), f'(x), and f"(x), respectively. Note: Input U, infinity, and -infinity for union, co, and -oo, 18. (1 point) respectively. If there are multiple answers, separate them by et Consider the function f (x) = commas. If there is no answer, input none. 8 tex' 16. (1 point) Consider the function f (x) = 2x + 9. (a) f'(x) = (b) f is increasing for x E (a) Find all critical numbers c off. c=- (c) f is decreasing for x E (b) f is concave up for x E_ (d) The local minima of f occur at x =. (c) f is concave down for x E. (e) The local maxima of f occur at x =_ (d) Using the 2nd derivative test, the local maxima of f oc- cur at x = (f ) f" ( xx ) =_ (e) Using the 2nd derivative test, the local minima of f oc- (g) f is concave up for x E_ cur at x = (h) f is concave down for x E. Note: Input U, infinity, and -infinity for union, co, and -co, respectively. If there are multiple answers, separate them by (i) The inflection points of f occur at x =. commas. If there is no answer, input none. Note: Input U, infinity, and -infinity for union, co, and -co, 17. (1 point) Suppose g(x) = (f(x))2 where f is positive and respectively. If there are multiple answers, separate them by concave up for all x E I. commas. If there is no answer, input none