Introduction
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Now we will see some Numerical methods and calculus questions with their solutions and proper explanation.
Questions with Solutions
1. What is the value of the following limit?
Solution: This is 0/0 form, by applying the L'Hospital rule.
2. Which of the following functions is/are increasing everywhere in [0,1]?
Solution) III only
3. What is the value of the following limit?
Solution) We will get 0/0 form when x = 3, so we can apply the L'Hospital rule.
4. Consider the equation , a quadratic equation with coefficients in a base b coefficients. This equation has two solutions in the same base b: x=5 and x=6. Then b is equal to =
Solution) 8
5. If , then the constants R and S are respectively.
Solution) 4/π and 0
Now, integrate equation (1) w.r.t. x, we get,
6. Let g(x)=f′(x) be the derivative of f(x), and f(x)=f′(x) be the polynomial. If (f(x)+f(-x)) has a degree of 10, then (g(x)-g(-x)) has a degree of =
Solution) 9
7. In the plot below, choose the most acceptable equation for the function represented as a thick line.
Solution) x = - (y - |y|)
8. The value of the double integral given below is -
Solution) ½
9. What is the value of the following limit?
Solution) ∞
10. If the slope of the tangent is -2x/y at every point of a certain curve, then the curve is
Solution) An ellipse
11. In the interval [-1,1] the function y=|x| is
Solution) Continuous but not differentiable
The function y = | x | in the interval [-1, 1 ] is
| x | is continuous and differentiable everywhere except at x = 0, where it is continuous but not differentiable.
Since [-1, 1] contains 0, in this interval it is continuous but not differentiable.
12. The formula for computing an approximation for a function f's second derivative at a location x0 is
Solution)
13. What is the value of the following integral?
Solution) (π/8) + (¼)
14. What is the value of the following limit?
Solution) 0
15. Take a look at these two statements about the function f(x)=|x|:
For all real x values, P.f(x) is continuous.
For any real x values, Q.f(x) is differentiable.
Which of the following statements is correct?
Solution) P is true and Q is false
f(x)=∣x∣. Here, for all values of x,f(x) exists. Therefore, it is continuous for all real values of x.
At x=0,f(x) is not differentiable. Because if we take the left hand limit here, it is negative while the right hand limit is positive making
LHL≠RHL
16. What is the value of the following limit?
Solution) 1
17. What is the value of the following double integral?
Solution) 13.5
18. What is the value of the following limit?
Solution) 1
19. What is the value of the following limit?
Solution) e-2
20. In the interval x∈[π/4,7π/4], consider the function f(x)=sin(x). The number and location(s) of this function's local minima are
Solution) One, at (3π/2)
21. The function f is defined at the following points-
Using the trapezoidal rule computes the value of the following integral-
Solution) 9.045
22. Which of the following functions is continuous at x=3?
Solution) (a)
23. Let the function f be
Where θ∈[π/6,π/3] and f(θ) denote the derivative of f with respect to θ. Which of the following statements is/are TRUE?
I) There exists θ∈[π/6,π/3] such that f(θ) =0.
II) There exists θ∈[π/6,π/3] such that f(θ) ≠0.
Solution) Both I and II
24. What is the value of k in the following integral?
Solution) 4
25. A function f(x) is linear, which has the value of 29 at x=−2 and 39 at x=3. Then find its value at x=5
Solution) 43
26. What is the value of the following limit?
Solution) 1
27. What is the value of the following limit?
Solution) 1