Applied Mathematics for A.I. Systems - Science & Maths Assignment Help

Assignment Task

Assignment: Applied Mathematics for A.I. Systems

A) Multiple Choice: Read each question carefully, then choose the best answer and write the letter of your choice in the space provided. If you think the correct answer is not among the choices then write E.

1.Simplify the following expression, b8(2b)4
A) 4b12 B) 2b12
C) 8b12 D) 16b12

2. When limx??fx=L, the line y = L is called
A) horizontal asymptote B) vertical asymptote
C) limit D) Continuous

3. What is the derivative of fx= x?
A) x1/2 B) x2
C) 82x D) 12x

4. The derivative of f(g(x)) is
A) 0 B) f’(g(x))(g’(x))
C) f’g + g’f D) f’(g’(x))

5. Another word for domain is the ______ inputs.
A) rangeB) dependent
C) relationD) independent

6. The Chain rule states:
A) dydx= dydududx B) dydx= dxdududyC) dxdy= dydududx.D) dydx= dududydx

7. The value f(c) is called the _________if f(c) ? f(x) for all x in domain.
A) absolute minimumB) absolute maximum
C) global minimumD) local maximum

8. The second derivative test says if f'c=0 and f''c>0, then f has a ______ at c.
A) local minimum B) local maximum
C) global maximumD) not enough information

9. If f and g are both differentiable, then the derivative of f(x) + g(x) is equal to
A) fg’ + gf’B) g’f + f’gC) neither D) f’(x) + g’(x)

10. The function f(x, y) = ax2 + x + c is called a _____ function.
A) cubic B) quadratic
C) bimodal polynomial D) linear

11. The function fx,y=x4+5x3y2+6xy4-7y+6 is an example of a _______.
A) polynomialfunctionB) rational function
C) quadratic functionD) none of the above

12. The first partial derivative with respect to y is for the function fx,y=x4+5xy3A) x4 + 15xy3B) 4x3 + 5y3
C) 4x3 + 15x3D) 15xy2

13. What is a function, f, of three variables?
A) A rule that assigns an ordered triple (x, y, z) in its domain a unique real number
B) A rule that assigns an ordered pair (x,y) in its domain a unique real number
C) Two of these
D) A surface level f(x, y) = k

B) True /False Read each statement carefully. On the space provided, write TRUE if the statement is true and write FALSE if the statement is false.
1. If fx= x2-xx-1 and gx=x is it true that f=g?
2. The slope of the line segment joining the points (1,-3) and (5, 7) is 5/2.
3. The order of the polynomial P(x) = x6 + xy + y is 2.
4. If f is a function, then f(s + t) = f(s) + f(t).

5. A function is differential when there is a kink at the point a.

6. If a function is continuous at a, then it is differentiable at a.
7. The derivative of the product is the product of the derivatives.
8. Functions that are open on an interval always contain extreme values.
9. If the first derivative at 0 = 0, (f'0=0), then the function must have a local max or min at 0.
10. The function f(x) = |x| is continuous at every point.

C) Problems: Solve the following application problems. Show all your work and box your final answers.
1. Evaluate f(0), f(2), f(-3) for the piecewise defined function. Is the function differentiable at -1?

2. For f(x) = 3x + 5 and g(x) = x2 + x; compute f(g(x)), g(f(x)), f(f(x)), g(g(x))
3. Evaluate the limit: limx?-1(x4-3x)(x2+5x+3)4. Compute the derivative of f(x) = (1 + x + x2)99
5. Find local maximum and minimum values of fx= 2x3-9x2+12x-36. Find the first partial derivatives for the function fx,y=ax+bycx+dy7. Suppose (0,2) is a critical point of a function g with continuous second derivatives. What can you say about g? gxx0,2=-1, gxy0,2=6, gyy0,2=18. Find the local maximum, minimum values and saddle points of the function. If you desire, please plot this with graphing software (matplotlib, grapher, etc).
fx,y=x2+y4+2xy

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