Calculus
Q1.
Suppose that f: \mathbb{R} \rightarrow \mathbb{R} is a continuous function on the interval [-3,3] and a differentiable function in the interval (-3,3) such that for every x in the interval, f'(x) \leq 2. If f(-3) =7, then f(3) is at most __________Q2.
The value of the definite integral \int_{-3}^{3} \int_{-2}^{2} \int_{-1}^{1} (4x^2y-z^3)dzdydx is _____. (Rounded off to the nearest integer)Q4.
The value of the following limit is _____ \lim_{x \to 0+} \frac{ \sqrt{x}}{1-e^{2 \sqrt{x}}}Q6.
For two n-dimensional real vectors P and Q, the operation s(P,Q) is defined as follows: s(P,Q) = \displaystyle \sum_{i=1}^n (P[i] \cdot Q[i]) Let \mathcal{L} be a set of 10-dimensional non-zero real vectors such that for every pair of distinct vectors P,Q \in \mathcal{L}, s(P,Q)=0. What is the maximum cardinality possible for the set \mathcal{L}?Q7.
Consider the functions I. e^{-x} II. x^2-\sin x III. \sqrt{x^3+1} Which of the above functions is/are increasing everywhere in [0,1] ?Q8.
Consider the following expression.\lim_{x\rightarrow-3}\frac{\sqrt{2x+22}-4}{x+3} The value of the above expression (rounded to 2 decimal places) is ___________.Q9.
The value of \int_{0}^{\frac{\pi }{4}}x\cos (x^{2})dx correct to three decimal places (assuming that \pi = 3.14 ) isQ10.
Let f(x)=x^3+15x^2-33x-36 be a real-valued function. Which of the following statements is/are TRUE?