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test 1 - Math l550 SecZO TEST 1 Name Note Please show your...

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Unformatted text preview: Math l550, SecZO TEST 1 Name: Note: Please show your steps for the problems with space. Points will be given basing on reasonable deductions. 1. (10 pts) f (x) = 5x2 +1 . Estimate the average ROC over [0, 2], and the instantaneous ROC at the point x = —l by numerical investigation. Round your answer to the first decimal place. - 2. , l .2“- Ht 244 Egg) ., limo) : W ; [N a ,2: [J . , , 2, o z '- Z a; ’ -l“’le 2. (10 pts) The function f (x) has the following graph. Evaluate each of the following expressions. Note: Write “DNE” if the limit does not exist or is not defined. f(0)= -l x—>O+ ——~——— 3153 fix} = Q ling f(x) = w u 3. (10 pts) Determine the point at which the function f (x) = i— is discontinuous, and state the type of x — 1 discontinuity: removable, jump, infinite, or none of these. Please explain by details. A’C ,‘in, @ gm {9 itndgfmai ., (x IE—lgit ,.—~ m Cd; A?! Cm “‘7'" "-1 *‘XJ 5;:- 3’ dau— in ’ , - . A i" "‘s’ll‘ani/L km 1 00 “ i u #(XlNiS Mimrit“ QSQLG ‘ [if . f:— -: + K X‘ai-i- 9” l +0 4. (10 pts) For what value of C is the function defined below continuous on (“06: 00)? (3324-1, x<6, mtg—1, 3326. mm) a: To soive the problem, just follow the hints. Solution: lim f(x) = éC 'i' i x—)6— ; lim f (x) = 36 C‘ ’- i . x—)6+ If f (x) is continuous, especially at x = 6, lim f(x) = lim f(x). x—>6— x-—)6+ Therefore, we can get an equation with respect to c: [”04” = 35c _) :9 13:0: L 1 Solving the equation, we get the value of c: A; 5. (20 pts) Evaluate the limit: (Note: If there is no limit, write DNE and explain.) lim ”CZ—25 1, 11m( 1 _ 1 ) , ._ a HS x—5 ' ) H3+ x—3 lx—3l ' (2.93 Jim” I"; 20 In C1§SQ(X+C;> ) . _ ”1,5! : 3 1‘ :3 z: JAM IN: 3: an) Bill? .4- X’L—3> _ 91-9.? — - 195+ 2 20 : o 6. (20 pts) Evaluate the limit: a) 11?}7‘2 005(6/3‘) (Use the Squeeze Theorem) b) lingw x— x—) x Z A ” 2 ‘ ' 3:1“) ~51 it 7c rls e/ch/Ec)s \ : Man (“(31) LEM __,_,,, ., I 190 let) 1/ 90 4534“ 1‘ dead- [1) é 7C ' . :1 (W amen) . r; :2- Win: 5’ 1 . [gm (#1:— , o 2 - ac"? a Q S .QCn/i 1,1 T 0 )L 7C By 4kg % 1466,33 MW Liv“ l La5Cé/ 1)}: 7. (10 pts) Find the derivative of the function f (x) 2362‘ 5+ Sex by formulas. “35 4%,): 27—“) x + 58 32.; x :..§1, + .5 er to deduce the derivative of f(X+h)-f(x) h 8. (10 pts) Use the definition of derivative f '(x) = £133 1- (1th). #3 \ + _ the function f (x):1 - MW" (X h) A” —~. x “— 11‘70 6‘ («i {(XHDWQU w 8L of a) : t3; 5 .m MW. 1 f” “Em/“50 lib [Ax an ‘ \< M .. ~ : vkéw 4’: ‘3‘ <k+ttfi< “A R “LL 0 X414 — f 'x “ l l X. _ J, .. V” m . 5. Sim l mu >< X h «we WM LL90 LL :: ~— -'~ ...
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