10 determine whether the given function satisfies the

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10. Determine whether the given function satisfies the conditions of the Mean Value Theorem on the given interval. If so, find all numbers c that satisfy the conclusion of the theorem. (a) f ( x ) = 2 x 2 - 3 x + 7, on x [0 , 2] (b) f ( x ) = 2 ln x , on x [1 , 4] (c) f ( x ) = tan 2 ( πx ), on x [0 , 1] (d) f ( x ) = 1 - 3 x 2 , on x [0 , 1] 11. For the given function find the following: (i) critical points; (ii) intervals of increase and intervals of decrease; (iii) points of local maximum and local minimum. (Note: If the domain (typically, an interval) is not provided, start by finding the domain of the gvien function — you’ll need it!) (a) f ( x ) = x 3 - 3 x 2 - 8 (b) f ( x ) = 3 x - x 3 (c) f ( x ) = 3 x 2 - 36 (d) f ( x ) = x 2 x 2 + 1 (e) f ( x ) = x 2 x 2 - 1 (f) f ( x ) = x 3 e - 5 x (g) f ( x ) = 3 x - 6 arctan x (h) f ( x ) = 3 sin x - cos x , on x [0 , 2 π ] 12. For the given function, find the following: (i) the intervals where its graph is concave up and where it is concave down; (ii) points of inflection. (Note: If the domain (typically, an interval) is not provided, start by finding the domain of the gvien function — you’ll need it!) (a) f ( x ) = x 4 + 2 x 3 (b) f ( x ) = x 2 x 2 + 4 (c) f ( x ) = 2 x 2 - 3 x + ln x (d) f ( x ) = ( x + 7) e x (e) f ( x ) = 3 x - 6 arctan x (f) f ( x ) = 2 x 2 + 8 cos x , on x [0 , 2 π ] 13. Find the given limit.
MATH-2211: Calculus of One Variable I Study Guide for Test 3 (out of 4) (g) lim x 3 ln(7 - 2 x ) arctan(3 - x ) (h) lim θ 0 arcsin θ 2 θ (i) lim θ 0 arctan θ 3 θ (j) lim x + e x 1 + ln x (k) lim t + (ln t ) 2 t (l) lim x π cos x + 2 - π/x sin 2 x (m) lim x 1 π - 4 arctan x x 3 - 1 (n) lim x 1 - (1 - x ) tan πx 2 (o) lim x 2 + ( x - 2) 5 ln( x - 2) (p) lim x 0 (1 + 2 x ) 3 csc(2 x ) (q) lim x 0 (1 + 4 x ) 3 /x (r) lim x 0 ( e 5 x + x ) 2 /x 14. Consider the function f ( x ) = x 3 + 3 x 2 . (a) Find the domain of the function. (b) Find all its x - and y -intercepts. (c) Is this function even or odd (or neither)? (d) Find the horizontal and vertical asymptotes of the graph (if any). Determine the behavior of the graph of f as x → ±∞ or as x approaches the endpoint(s) of the domain (whichever is applicable). (e) Find the critical points, the intervals on which f is increasing or decreasing, and all local extreme values of f . (f) Find the intervals where f is concave up or concave down and all inflection points. (g) Use the information you have found above to sketch the graph of y = f ( x ). 15. Consider the function f ( x ) = x 4 9 - 2 x 2 . (a) Find the domain of the function. (b) Find all its x - and y -intercepts. (c) Is this function even or odd (or neither)? (d) Find the horizontal and vertical asymptotes of the graph (if any). Determine the behavior of the graph of f as x → ±∞ or as x approaches the endpoint(s) of the domain (whichever is applicable). (e) Find the critical points, the intervals on which f is increasing or decreasing, and all local extreme values of f . (f) Find the intervals where f is concave up or concave down and all inflection points.

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