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Mean Value Thereom Solutions

# Mean Value Thereom Solutions - CALCULUS WORKSHEET ON THE...

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Unformatted text preview: CALCULUS WORKSHEET ON THE MEAN VALUE THEOREM 1. Use the graph of f to estimate the numbers in [0, 8] that satisfy the conclusion of the Mean Value Theorem. A {V /~3ﬁlf§‘“‘f 2. Use the graph of gto estimate the numbers in [0, 8] that satisfy the conclusion of the Mean Value Theorem. In problems 3 — 10, determine Whether f satisﬁes the hypotheses of the Mean Value Theorem on b _ the interval [0, b]. Ifit does, ﬁnd all numbers 6 in (a, [7) such that f ’(c) : —f—(—%——f—(—C—I—)— — a 3 f(:)= 3x2 +x— 4 Q[1,5] -/ it,“ £ (24)?“ Méfﬁgﬁ Maﬁa; ﬂair m j 10. f(x) : 23inx+sin(2x) [0, 7r] . ‘ Jx+2, xSl ll. Sketch a graph ofthe function f If f(x): l 2 1 ' x , x> Show that f fails to satisfy the hypothesis of the Mean Value Theorem on the interval [~ 2, 2], but the conclusion of the theorem is still valid. as f ‘ A”? 54......” 5 If“? V7“ Answers 1. oz 1.5 and 5.5 2. c z 1,35, and 6.5 3. f does satisﬁes the MVT. c = 3 4. f is not differentiable at x = 0 so it does not satisfy the MV T. , 5. f does satisﬁes the MVT. c = 27—f— 4 6. f is not differentiable at x = 0 so it does not satisfy the MVT. 7. f does satisﬁes the MVT. c = -1 8. f does satisﬁes the MVT. c = 3% 9. f does satisﬁes the MVT. c = 2;— 10. f does satisﬁes the MVT. c = £— "9 D 11. f is not continuous at x = 1, but all of the values ——2 S x <1 on the left side of the piecewise function satisfy the conclusion of the MVT. 12. The distance is a continuous, differentiable ﬁinction. For the points (0, O) and (2, 159), the speed is 74.5 mph, which is greater than 65 mph. 13. The temperature is a continuous, differentiable function. For the points (0, O) and (20, 212), the rate of change is 10.6°F/sec. 14. The distance is a continuous, differentiable function. His average speed is 11.909 mph so he must have run at that speed at least once. Since his initial speed and ﬁnal speed are both 0 mph and the runner’s speed is continuous, by the Intermediate Value Theorem, his speed must have been 11.909 mph at least twice. CALCULUS WORKSHEET ON THE MEAN VALUE THEOREM 1. Use the graph of f to estimate the numbers in [0, 8] that satisfy the conclusion of the Mean Value Theorem. 2. Use the graph of g to estimate the numbers in [0, 8] that satisfy the conclusion of the Mean Value Theorem. X35 at? In problems 3 — 10, determine whether f satisﬁes the hypotheses of the Mean Value Theorem on b .. the interval [61, b]. Ifit does, ﬁnd all numbers 6 in (a, b) such that f ’(c) = w. — a 3. f(x)=3x2+x—4 [1,5] Woe: 9m»! z} “4% W 2/9 Cf; Q J ‘9 2:? . 7: 57: 5.f(x)=cosx—smx [—,—:| ”55" 9m» (@4ng 4/ x. 4 4 i “ﬁjw’pﬁﬁieﬁma‘om 9% song-sums...“ M > m»..\‘ 10. f(x):2sinx+sin(2x) [0,72] x+2, x31 ‘7 x‘ x>l .1 ll. Sketch a graph ofthe function f if f(x) :{ Show that f fails to satisfy the hypothesis of the Mean Value Theorem on the interval [- 2, 2], but the conclusion of the theorem is still valid. I " 4y , €- i; g / /{/§’j ( 5/7» 3 3‘ = 1 filter 5; 7' ”F“ "I r ’ 12. A trucker handed in a ticket at a toll booth showing that in 2 hours he had covered 159 miles on a toll road with speed limit 65 mph. The trucker was cited for speeding. Why? ”Lax iﬂﬁwtv ,: are“? «ewe? 13. It took 20 sec for the temperature to rise from O°F to 212°F when a thermometer was taken from a freezer and placed in boiling water. Explain why at some moment in that interval the mercury was rising at exactly 10.6°F/sec. 4% at“? ‘sggr , W 35,, w t M .3} vs» that! «a 14. A marathoner; ran the 26.2:mi New York City‘Myarathon in 22 h. Show that at least twice, the marathoner was running at exactly 11 mph. tr‘ﬁ‘» 3w / «a»! ' V A—r‘y 3Ww¢ t”? % A: ”law’wﬂ v) 5' gas,“ 9;: ”77” @3135 W ‘81: 5N; f“? y? S ources : 1) Calculators ~ Concepts and Calculators, Second Edition — Best, Carter, Crabtree 2) Calculus — GraphicaILNumericaL Algebraic, Third Edition — F inney, Demana, Waits, Kennedy ...
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