Solution : y = \(x^3 – 6x^2 + 12x + 5\) y’ = \(3x^2 – 12x + 12\) y” = \(6x – 12\) y” = 0 \(\implies\) 6x – 12 = 0 \(\implies\) x = 2 Since, y” = 0 at x = 2, Hence the point of inflection is 2. Similar Questions Prove that […]
Find the point of inflection for the curve y = \(x^3 – 6x^2 + 12x + 5\). Read More »
Solution : f(x) = \(3x^4 – 4x^3\) f'(x) = \(12x^3 – 12x^2\) f'(x) = \(12x^2(x – 1)\) Now, f”(x) = \(12(3x^2 – 2x)\) f”(x) = 12x(3x – 2) f”(x) = 0 \(\implies\) x = 0, 2/3 Here, f”(x) = 0 Thus, x = 0, 2/3 are the inflection points. Similar Questions Prove that the function
Find the inflection point of f(x) = \(3x^4 – 4x^3\). Read More »
Solution : The given parabola is of form \(y^2\) = 4ax. On comparing, we have 4a = 12 i.e a = 3. We know that the focal distance of any point (x, y) on \(y^2\) = 4ax is x + a. Let the given point on the parabola \(y^2\) = 12 x be (x, y).
Solution : Let y = cosx.sinx By using product rule in differentiation, \(dy\over dx\) = sinx(-sinx) + cosx.cosx \(dy\over dx\) = \(cos^2x – sin^2x\) = cos 2x Hence, the differentiation of cosx.sinx with respect to x is cos 2x. Questions for Practice What is the differentiation of \(e^{sinx}\) ? What is the differentiation of sin
What is the differentiation of cosx sinx ? Read More »
Solution : Let y = \(e^{sinx}\). Putting u = sinx , we get y = \(e^u\) and u = sinx \(\therefore\) \(dy\over du\) = \(e^u\) and \(du\over dx\) = cosx Now, \(dy\over dx\) = \(dy\over du\) \(\times\) \(du\over dx\) \(\implies\) \(dy\over dx\) = \(e^u\)cosx = \(e^{sinx}\)cosx Hence, the differentiation of \(e^{sinx}\) with respect to x
What is the differentiation of \(e^{sinx}\) ? Read More »
Solution : The differentiation of sin square x with respect to x is sin 2x. Explanation : We have, y = \(sin^2 x\) Differentiating by using chain rule, \(dy\over dx\) = 2 sin x cos x \(dy\over dx\) = sin 2x Hence, \(dy\over dx\) = sin 2x Questions for Practice What is the differentiation of
What is the differentiation of sin square x or \(sin^2x\) ? Read More »
Solution : We have, y = 1/sinx = cosecx By using differentiation formula of cosecx, \(dy\over dx\) = -cosecx.cotx Hence, the differentiation of 1/sinx = -cosecx.cotx Questions for Practice What is the differentiation of \(e^{sinx}\) ? What is the differentiation of sin square x or \(sin^2x\) ? What is the differentiation of cosx sinx ?
What is the differentiation of 1/sinx ? Read More »
Solution : We have, y = \(sin x^2\) Differentiating with respect to x by using chain rule, \(dy\over dx\) = \(cos x^2\).(2x) \(dy\over dx\) = 2x.\(cos x^2\) Hence, the differentiation of \(sin x^2\) with respect to x is 2x.\(cos x^2\) Questions for Practice What is the differentiation of \(e^{sinx}\) ? What is the differentiation of
What is the differentiation of \(sin x^2\) ? Read More »
Solution : We have, I = \(\int\) log cos x dx By using integraton by parts, I = \(\int\) 1.log cos x dx Taking log cos x as first function and 1 as second function. Then, I = log cos x \(\int\) 1 dx – \(\int\) { \({d\over dx}\) (log cos x) \(\int\) 1 dx
What is the integration of log cos x dx ? Read More »
Solution : We have, I = \(\int\) \(log {1\over x}\) dx I = \(\int\) \(log 1 – log x\) dx = \(\int\) (-log x) dx By using integration by parts formula, Let I = -(\(\int\) log x .1) dx where log x is the first function and 1 is the second function according to ilate
What is the integration of log 1/x ? Read More »
