Showing posts with label calculus. Show all posts
Showing posts with label calculus. Show all posts

Jul 18, 2009

Differential and Integral Calculus - Multiple Choice Questions - Part II

1. What is the primary difference between using anti-differentiation when finding a definite versus an indefinite integral?

Indefinite integrals don't have defined limits.

Definite integrals have defined limits.

The constant of integration, C.

There is no difference between definite and indefinite integrals.

2. If θ approaches zero, then limit of sin(θ)/θ is _____.

cos(θ)

0

1

This is indeterminate.

3. Let's say that f ''(k) = 0 @ (13, -2). What does this mean?

There is definitely an inflection point at that location.

There might be an inflection point at that location.

There definitely is not an inflection point at that location.

There's no way to tell without first knowing what the specific function is.

4. What is the one thing done in anti-differentiation that has no counterpart in differentiation?

Adding a constant C.

Subtracting a constant C.

Dividing the new exponent by a constant C.

Nothing, they are equally matched step by step.

5. What is a necessary condition for L'Hôpital's Rule to work?

The function must be determinate.

The function must be indeterminate.

The function must be inconsistent.

The function must possess at least three non-zero derivatives.

6. What does du equal in ∫2x(x2 + 1)5 dx?

2x

2u du

2x dx

5u4

7. What is the second step of performing anti-differentiation?

Divide the coefficient by the old exponential value.

Subtract the new exponential value from the coefficient.

Multiply the coefficient by the new exponential value.

Divide the coefficient by the new exponential value.

8. Which of the following is the best integration technique to use for ∫2x(x2 + 1)5 dx?

The product rule.

The chain rule.

The power rule.

The substitution rule.

9. ∫1/x dx =

Undefined because you cannot divide by zero.

loge(x)

ln(x)

ln(x) + C

10. What is the converted substitution form of ∫12x2 (x3 + 1)5 dx?

4∫u5 du

∫u5 du

¼ ∫u5 du

This cannot be solved by the substitution method.

To see Part - I, click here

Jul 11, 2009

Differential and Integral Calculus - Multiple Choice questions - Part I


1. If f(x) = 3x2, then F(x) =
   a) 6x
   b) x3
   c) x3 + C
   d) 6x + C

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2. The two types of errors that are related to differentials are:
    a) Human, Absolute.
    b) Absolute, Relative.
    c) Relative, Controllable.
    d) Controllable, Natural.

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3. Mathematically, what is a differential?
    a) A gear box on the back end of your car.
    b) A word used a lot on a popular medical television series.
    c) A method of directly relating how changes in an independent variable affect changes in a dependent variable.
    d) A method of directly relating how changes in a dependent variable affect changes in an independent variable.

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4. The 2nd derivative of a function at point P is 0, and concavity is positive for values to the right of P. What must the concavity be to the left of P for P to be an inflection point?
   a) The concavity must also be positive.
   b) The concavity must be negative.
   c) The concavity must be neutral (0).
   d) The concavity must be imaginary.

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5. At what value of q is the concavity of w(q) = -2, if w(q) = q4 - 16?
    a) At q = fourth root of 14.
    b)At q = 0.
    c) Never; w(q) is always concave down.
    d) Never; w(q) is always concave up.

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6. What is needed to fully determine an anti-differentiated function?
     a) A lot of luck.
     b) A boundary condition.
     c) What its value is at (0, 0).
     d) Its real world application.

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7. It has been determined that g(p) has a maximum at p = -47.6. What can be said of the function's concavity at that point?
   a) g ''(p) = 0
   b) g ''(p) > 0
   c) g ''(p) < 0
   d) There's no way to tell without first knowing what the specific function is.


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8. What are the values of C0 and C1 in d(t) = C1 + C0t - 16t2, if d(1) = 4 and v(2) = -65?
    a) C0 = -1, C1 = 21
    b) C0 = 1, C1 = -21
    c) C0 = -1, C1 = 19
    d) C0 = 0, C1 = 1

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9. G(d) was determined to be 3d + C; here, C is called:
     a) the constant of differentiation.
     b) the constant of anti-differentiation.
     c) the constant of integration.
     d) the constant of death and taxes.

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10. Does f(c) = (c + 2)3 − 2 have an inflection point? If so, where is it located?
     a) Yes, at (-2, -2).
     b) Yes, at (2, -2).
     c) Yes, at (8, -2).
     d) No.

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