By Mark McKibben

Designed for these looking support learning calculus in class - additionally important for adults trying to learn/re-learn calculus. A source for teachers supplementing their guide. 501 Calculus difficulties is helping clients arrange for educational assessments and construct problem-solving talents. in contrast to textbooks, complete resolution factors are supplied for all difficulties. contains: - Questions protecting all of unmarried variable calculus - universal calculus blunders - specified answers/fully labored recommendations - thesaurus and theorem checklist - entry to unfastened on-line try

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59. By definition, ( f f f )(2 x) = f ( f ( f (2x ))). Working from the inside outward, we first note that f (2 x ) = −(2x )2 = −4 x 2 . Then, f ( f (2x )) = f ( ) = − ( −4 x ) = −16x . And finally, f ( f ( f (2x))) = f ( −16x ) = − ( −16 x ) = −256 x . Thus, we conclude that ( f f f )(2x ) = f ( −4 x 2 2 2 4 2 4 4 8 = −256 x 8 . qxd 4/25/12 12:42 PM Page 47 501 Calculus Questions 60. , a square root) must be nonnegative if in the numerator of a fraction and strictly positive if in the denominator of a fraction.

A. [ 0, ∞ ) b. ( −∞,0 ] c. ޒ d. none of the above 84. If f (x ) = 1 , simplify 2x the expression f ( x + h) − f ( x ) , where h h ≠ 0. 85. Which of the following sequence of shifts would you perform in order to obtain the graph of f ( x) = ( x + 2)3 − – 3 from the graph of g (x ) = x 3 ? a. Shift the graph of g(x) up three units and then left two units. b. Shift the graph of g(x) down three units and then right two units. c. Shift the graph of g(x) up three units and then right two units. d. Shift the graph of g(x) down three units and then left two units.

Isolate the radical term on one side of the equation by dividing by 3, then square both sides to obtain 3 x = 18 4 ( 2xy 4 −1 3 4 x =6 x = 36 22. qxd 4/25/12 12:39 PM Page 22 501 Calculus Questions 23. Since the radical term is already isolated, we begin by squaring both sides, and subsequently moving all terms to the left side (so that the coefficient of the term with the largest exponent is positive). z = z + 12 z 2 = z + 12 z 2 − z − 12 = 0 Now, factor the expression on the left using the trinomial method to obtain the equivalent equation ( z − 4)(z + 3) = 0 Since the product of two real numbers is zero if and only if one of the numbers itself is zero, we conclude that at least one of ( z − 4) and (z + 3) is zero, so that the solutions of the factored equation are z = −3 and z = 4 .

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