Anna University Questions
Anna University Questions
PART A QUESTIONS
Represent of function
-
Sketch the graph of the function
\(f(x)= \begin{cases} x^2 & \text{if } -2 \leq x \leq 0 \\ 2-x & \text{if } 0 < x \leq 2 \end{cases}\).
[N/D-23-R-21]
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Sketch the graph of the function \(f(x)=2-0.4x\) and find the domain of the function.
[A/M-24-R-21]
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What is the effect of the first derivatives on the shape of the function?
[A/M-26-R-25]
-
Find the domain of the function \(f(x)=\frac{\sqrt{x+2}}{x-3}\).
[A/M-26-R-25]
-
Write the domain of the following functions:
- \(f(x)=\sqrt{x+2}\)
- \(g(x)=\frac{1}{x^{2}-x}\)
[A/M-26-R-21]
-
Find the domain of the function \(f(x)=\frac{2x^3-5}{x^2+x-6}\).
[JAN-22-A/M-22-R-21]
-
Find the domain of the function \(f(x)=\frac{1}{x^2-x}\).
[N/D-22-R-21]-CUR
Limit
-
Investigate the \(\lim_{x\rightarrow0}\sin\left(\frac{\pi}{x}\right)\).
[N/D-25-R-21]
-
Find: \(\lim_{x\rightarrow1}\frac{x^{2}-1}{x-1}\)
[A/M-26-R-21]
-
Evaluate: \(\lim_{x\to-2/3}(9x^{2}-12x-4)\).
[N/D-25-R-25]
-
Find the value of \(\lim_{x \to 1} \frac{x-1}{x^2-1}\).
[A/M-25-R-21]
-
Evaluate the limit \(\lim_{x\to 1} \frac{x^2-4x}{x^2-3x-4}\).
[JAN-22-R-21]
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Evaluate: \(\lim_{x\to 5} (2x^2-3x+4)\).
[A/M-22-R-21]
-
Prove that \(\lim_{x\to 0 } \frac{\vert x \vert }{x}\) does not exist.
[N/D-22-R-21]-CUR
CONTINUITY
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Is the function \(f(x)=\frac{x^{2}-1}{x-1}\) continuous at \(x=1\)? Justify.
[N/D-25-R-21]
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For what values of constant \(c\) is the function \(f\) continuous on \((-\infty,\infty)\),
\(f(x)=\begin{cases} cx^2+2x, & x<2 \\ x^3-cx, & x\geq 2 \end{cases}\).
[N/D-22-R-21]-ARR
Derivative
-
Find \(\frac{dy}{dx}\) if \(x^{2/3}+y^{2/3}=a^{2/3}\).
[A/M-26-R-25]
-
Find \(\frac{dy}{dx}\) if \(ax^{2}+2hxy+by^{2}=c\).
[N/D-25-R-25]
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If \(x = a(\cos t + t \sin t)\) and \(y = a(\sin t - t \cos t)\), find \(\frac{d^2y}{dx^2}\).
[A/M-25-R-21]
-
Find the slope of the circle \(x^2+y^2=25\) at \((3,-4)\).
[N/D-22-R-21]-ARR
-
If \(y=x\log\left( \frac{x-1}{x+1} \right)\) then find \(\frac{dy}{dx}\).
[A/M-23-R-21]
-
Differentiate \(y=x\tan\sqrt{x}\) with respect to \(x\).
[A/M-24-R-21]
-
The equation of motion of a particle is given by \(s=2t^3-5t^2+3t+4\) where \(s\) is measured in meters and \(t\) in seconds. Find the velocity and acceleration as a function of time.
[N/D-23-R-21]
Maxima & Points
-
Find the point of inflection of \(f(x)=x^3-9x^2+7x-6\).
[A/M-23-R-21]
-
What is meant by saddle point?
[N/D-25-R-25]
✔ ———————————— ✔
PART B QUESTIONS
CONTINUITY
-
Find the values of \(a\) and \(b\) so that
\(f(x)=\begin{cases}ax+b, & x\le1\\ x^{2}, & 12\end{cases}\)
is continuous at \(x=1\) and \(x=2\).
[A/M-26-R-25]
-
For what value of \(a\) and \(b\) is
\(f(x)=\begin{cases} ax+2b, & x \leq 0 \\ x^2+3a-b, & 02 \end{cases}\)
continuous at every \(x\).
▶ Video Solution
[N/D-25-R-25]
-
For what values of \(a\) and \(b\), is
\(f(x)=\begin{cases} -2, & x\leq -1 \\ ax-b, & -1
[A/M-22-R-21]
Show that the function \(f(x)\) is continuous on \((-\infty,\infty)\),
\(f(x)=\begin{cases} 1-x^2, & x\leq 1 \\ \log x, & x\geq 1 \end{cases}\).
[N/D-22-R-21]-CUR
Find the value of \(a\) and \(b\) that make \(f\) continuous on \((-\infty,\infty)\) if
\(f(x)=\begin{cases} \frac{x^3-8}{x-2}, & \text{if } x<2 \\ ax^3-bx+3, & 2\leq x <3 \\ 2x-a+b, & x \geq 3 \end{cases}\).
▶ Video Solution
[A/M-23-R-21]
Let \(f(x)=\begin{cases} \sqrt{-x}, & \text{if } x<0 \\ 3-x, & \text{if } 0 \leq x \leq 3 \\ (x-3)^2, & \text{if } x>3 \end{cases}\). Evaluate each of the following limits, if they exist:
(i) \(\lim_{x\to 0^{-}} f(x)\), (ii) \(\lim_{x\to 0^{+}} f(x)\), (iii) \(\lim_{x\to 3^{-}} f(x)\),
(iv) \(\lim_{x\to 3^{+}} f(x)\), (v) \(\lim_{x\to 0} f(x)\), (vi) \(\lim_{x\to 3} f(x)\).
Also find where \(f(x)\) is continuous.
[N/D-23-R-21]
LIMIT OF THE FUNCTION
Find the value of \(\lim_{x\to 2 } \left[ \frac{x^2-2}{x^3-3x+5} \right]^2\).
[A/M-24-R-21]
Find \(\lim_{x\to 0} \frac{\tan x - x}{x^3}\).
[A/M-25-R-21]
Water runs into a conical tank at the rate of \(9\text{ ft}^3/\text{min}\). The tank stands point down and has a height of \(10\text{ ft}\) and a base radius of \(5\text{ ft}\). How fast is the water level rising when the water is \(6\text{ ft}\) deep?
[A/M-25-R-21]
DERIVATIVE
Using derivative, perform the following:
- \(\frac{d}{dx}\ln\left(\frac{x+1}{x-1}\right)\)
- Find \(y'\) if \(\sin(x+y)=y^{2}\cos x\).
[N/D-25-R-21]
Differentiate each of the following functions. Show all work and specify the differentiation rules applied at each step:
- \(f(x)=x^{8}+12x^{5}-4x^{4}+10x^{3}+6x-5\)
- \(f(x)=(a+bx)\sqrt{x}\)
- \(f(x)=\frac{e^{x}}{(1+x^{2})}\)
- \(f(x)=x \, e^{x}\sin x\)
[N/D-25-R-21]
Find the derivative of \(g(t)=\tan(5-\sin(2t))\).
[A/M-26-R-21]
Find the \(n^{\text{th}}\) derivative of \(f(x)=xe^x\).
Differentiate \(F(t)=\frac{t^2}{\sqrt{t^3+1}}\).
[N/D-23-R-21]
Find an equation of the tangent and normal lines to the given curve at specified point \(f(x) = \frac{x^2 - 1}{x^2 + x + 1}\), at \((1, 0)\).
[A/M-25-R-21]
If \(x^2+y^2=25\), then find \(\frac{dy}{dx}\) and also find an equation of the tangent line to the curve \(x^2+y^2=25\) at the point \((3,4)\).
[JAN-22-R-21]
Find the equation of the tangent line to the curve \(y=\frac{e^x}{(1+x^2)}\) at the point \((1,e/2)\).
[N/D-22-A/M-24-R-21]-CUR
If \(f(x)=xe^x\) then find \(f'(x)\). Also find the n-th derivative \(f^{(n)}(x)\).
[JAN-22-R-21]
Differentiate the function \(f(x)=\frac{\sec x}{1+\tan x }\). For what value of \(x\), the graph of \(f(x)\) has a horizontal tangent.
[JAN-22-R-21]
Find the differential coefficients of \(\frac{(a-x)^2(b-x)^3}{(c-2x)^3}\).
[A/M-22-R-21]
Evaluate (1) \(\frac{d}{dx}(3x^5 \log x )\) and (2) \(\frac{d}{dx} \left( \frac{x^3}{3x-2}\right)\).
[A/M-22-R-21]
Find \(y''\) if \(x^4+y^4=16\).
[N/D-22-R-21]-ARR
Differentiate \(y=(2x+1)^5(x^3-x+1)^4\).
[N/D-22-R-21]-ARR
Find \(\frac{dy}{dx}\) if \(y=x^2e^{2x}(x^2+1)^4\).
[A/M-23-R-21]
If \(x^y=y^x\) prove that \(\frac{dy}{dx}=\frac{y(y-x\log y )}{x(x-y\log x )}\) using implicit differentiation.
[A/M-23-R-21]
Use logarithmic differentiation to differentiate \(y=\frac{x^{3/2}\sqrt{x^2+1}}{(3x+2)^5 }\).
[N/D-23-R-21]
Slope
Find the slope of the circle \(x^{2}+y^{2}=25\) at the point \((3, -4)\).
[A/M-26-R-21]
MAXIMA AND MINIMA FOR SINGLE VARIABLE
Find the maximum and minimum values of \(f(x)=2x^{3}-9x^{2}+12x-5\).
[A/M-26-R-25]
Find the maximum and minimum values of \(f(x) = 3x^4 +4x^3 -12x^2 +12\).
[N/D-25-R-25]
Find the maximum and minimum values of \(2x^3-3x^2-36x+10\).
[A/M-22-R-21]
Show that \(\sin x (1+\cos x )\) is maximum when \(x=\frac{\pi}{3}\).
[A/M-23-R-21]
Find the local maximum and minimum for the function of the curve \(y=x^4-4x^3\).
[N/D-22-N/D-23-R-21]-CUR
Find the local maxima, local minima, and points of inflection (if any) for the function \(f(x)=x^{3}-6x^{2}+9x+1\). Also, determine the nature of each critical point using the second derivative test.
[A/M-26-R-21]
Find the intervals on which \(f(x)=-x^3+12x+5; \quad -3\leq x \leq 3\) is increasing and decreasing. Where does the function assume extreme values? What are those values?
[N/D-22-R-21]-ARR
Find the maximum and minimum values of \(f(x) = 3x^4 - 2x^3 - 6x^2 + 6x + 1\) in the interval \((0,2)\).
[A/M-25-R-21]
Find the absolute maximum and minimum values of the function \(f(x)=x^{3}-3x^{2}+1\) where \(\frac{1}{2}\le x\le4\).
[N/D-25-R-25]
Find the absolute maximum and absolute minimum values of the function \(f(x)=3x^4-4x^3-12x^2+1\) on the interval \([-2,3]\).
[JAN-22-R-21]
Find the absolute maximum and absolute minimum values of the function \(f(x)=\log\left[ x^2+x+1\right]\) in the interval \([-1,1]\).
[A/M-24-N/D(CUR)-22-R-21]
MEAN VALUE THEOREM
State mean value theorem and verify it for the function \(f(x)=x^{2}+2x-3\) in the interval \([0, 2]\).
[A/M-26-R-25]
State mean value theorem and verify it for the function \(f(x)=x^{2}-4x-3\) in the interval \([1, 4]\).
[N/D-25-R-25]
Verify Rolle's theorem for \(f(x)=x^3-x^2+6x+2\) on \([0,3]\).
Verify Lagrange's MVT for \(f(x)=2x^2-3x+1\) on \([0,2]\).
Verify Lagrange's MVT for \(f(x)=x^2+2x-1\) on \([0,1]\).
Verify Lagrange's MVT for \(f(x)=\sqrt{x-1}\) on \([1,3]\).
✔ ———————————— ✔
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