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Calculus

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Published in: Mathematics
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7 Most common mistakes in calculus

Brijesh S / Mumbai

10 years of teaching experience

Qualification: B.Tech/B.E. (Mumbai University - 2010)

Teaches: Biology, Chemistry, Algebra, English, Marathi, Mathematics, Science, Special Education, Vedic Maths

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  1. 7 MOST COMMON MISTAKES IN CALCULUS L hospitals rule 1) Determine, lim (axln(bx)) The Mistakes, lim (axln(bx)) lim aln(bx) + ax • — b bx (The notation "I'H" above the equals sign indicates a step at which l'Höpital's rule is claimed to be used) The first mistake is that the student has misunderstood L 'Hospital's rule to apply to a products. L 'Hospital's Rule tells us that if we have an indeterminate form 0/0 or 00/00 all we need to do is differentiate the numerator and differentiate the denominator and then take the limit. Thus the first step is to rearrange the limit expression into the form of an indeterminate form. 1 In(bx) PH lim (axln(bx)) lim x 1 1 2 1 a The second mistake shows one way to correctly rearrange the expression; the mistake comes only at the end of the calculation when the final rational expression is simplified incorrectly. A Correct Solution lim (axln(bx)) — 1 In(bx) lim 1 1 2 lim 1 •(—ax2) lim (—ax) 0
  2. The Mistakes, i. f (x) = lim 2) Use the limit definition of the derivative to find the derivative of f'(x) = lim h f(x) = I—x2 I—x2 h I x2+2xh + h2 I x2 1—x2+2xh+h2+ I x2 h 1 x2+2xh+ h2— (1 —x2) 1 —x2+2xh + h2+ I —x 1 —x2+2xh 4- h2+ 1 —x2 h(2x + h) x I—x2 12) 2x 2x I—x24- 1—x2 All three of the attempted solutions have errors in distributing a negative sign over a factor. The expression (x+h)2 has a negative sign in front of it, so when that expression is expanded every terms must have its sign changed. The first and second mistakes fail to do this. ii. 1 (x+h)2 I—x2 h h 12) h 1 2—2xh+hZ h —2-2x The second mistake is compounded with two further errors. First, the limit expression is multiplied by an expression not equal to 1, which therefore changes the value of the expression and the limit. Second, for no apparent reason the sign is reversed on one of the "1 "s in the denominator. Third, h is cancelled from the numerator and denominator although h is not a common factor of the numerator.
  3. In the third mistake the sign error occurs in the step after the rationalization (multiplying the numerator and denominator by the "conjugate expression " of the numerator); the second term is -(1-x2) = -1 + x2. Once again, h is cancelled from numerator and denominator without being a common factor of the numerator. iii. f(x+h) h 1 x2—2xh —i h2 h 1 —x2 —i 2xh — h2 h 1 x2 1—x2 1 x2 x2 2xh Il 2x2 — 2xh h2 — 12 —i 2.xh — h2 -F 2X2 212 12 A Correct Solution f(x+h) f(x) h—sCJ h I—x2 f (x) lim h 2x h | 1—x2 x2 h( 2x h) — 12 2xh — h2 + —2x —x I—x2+ 1 —xo — 2 1—x2 Use extreme care with negative signs, especially in problems such as these.
  4. Differentiation of exponential functions 3) Find, d The Mistakes, d Here the student has applied power rule and an exponential function is not a power function, so the power rule cannot be applied to find its derivative. A power function has independent variable in its base (in our case the independent variable is 't') while an exponential function has the independent variable in the exponent of the function (in our case its et). So in this case use the formula for the derivative of an exponential function. The corrections, d dt 4) Find d (2) dx The Mistakes, d dx —1 This is the similar case as above. 2x is an exponential function and power rule cannot be applied to find its derivative. A power function has independent variable in its base (in our case the independent variable is 'x') while an exponential function has the independent variable in the exponent of the function (in our case its 2x). So in this case use the formula for the derivative of an exponential function. The Correction, d (r) — In(2) dx
  5. Differentiation of Hyperbolic functions 5) Find, d — cosh(x) dx The Mistakes, —COsh(x) = — sinh(x) Here the student has misconception between trigonometric function and hyperbolic function. If it would have been just a trigonometric function then the derivative of cosx would have been —sinx. But the derivatives of the hyperbolic functions don 't agree with the corresponding trigonometric derivatives in sign in all cases. d er+e — cosh(x) —r 2 Here the student has confused the definition of hyperbolic cosine (in terms of exponentials) with the derivative of hyperbolic cosine. Note: Hyperbolic cosine of x = cosh x = (ex + e-X)/2 Correction, d —cosh(x) sinh(x) dx Basic derivative formulas should be learned. For hyperbolic cosine there are two choices: memorize the correct derivative or derive the derivative using the definition of hyperbolic cosine whenever this derivative is needed: d 1 cosh(x) dx dx 2 Note: Hyperbolic sine of x = Sinh x d dx dx 1 — — sinh(x)
  6. Tangent Lines 6) Find the equation of the tangent line to f(x) - At x=2 The Mistake, f (x) x3 f (x) 3x2i Since also f (2) 8, the tangent line is y— 8 3x2(x— 2), or y 3x3 — 6x2 + 8. The derivative off(x) isf'(x) from which the slope of a tangent line can be computed at a point x=a by evaluating f '(x) at x=a, that is, by computing m=f'(a). The result is a number, which then can be used to find the equation of the tangent line at x=a using slope-point form for a line, in this situation: y -f(a) = m(x-a). The Correction, Therefore the slope of the tangent line at x = 2 is , f(2) —12i Since also f ( 2) the tangent line is y — 8 —12(x 2) } or y --12x— 16.
  7. Quotient Rule 7) Find, d dz The Mistakes, 1 d 3z2+ 1 dz z4—5 3z2+ 1 (3z2+ (z4 — Here bracket should have been used around the factor z4-5; not using them meant that the 6zfactor was not distributed over the whole factor z4-5. Correction, d (3z2-F IN 6z(z4 —5) "—e (3?+1) dzkz4—5)— d 3z2+1 LAI dz z4—5 —6Z5 — 4Z3 '—e 30Z (z4 (z4 — 5) (6z) 6z5+4z3 + 30z (z4 — Here the numerator terms were reversed; this mistake changes the sign of the result. Take care to get the order of the terms correct in the numerator when using the quotient rule. Correction, d 3z2+ 1 h 6z(z4 dz z4—5)— —6? — 4? — 30z (z4

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