For the car given, t₁=1 and t₂=5. The values of f at these points are f(t₁) = f(1) = 1³+1²+1 = 3,
f(t₂) = f(5) = 5³ + 5² +5 = 155. So, the average velocity is equal to (155 - 1)/(5 - 1) = 154/4 = 38,5 metres/seconds. The answer is B.

If f(x,y)=3x²y+e^xy+lny

?f /?x=6xy+ye^xy

^{?²f /?x²=6y+y²}e^xy

The interval consisting of the elements of both intervals (-1/3, 4] and [-1, 7/2) is [0, 7/2). The answer is A.

The price of a product at the intersection point, provided that it is in the first quadrant, is called market price.  

When we put it on the demand or supply equations, we find market price as 1700. 

From the product rule for limits, we have 

. The answer is B.

grad f (x, y, z) = (?f / ?x , ?f / ?y , ?f / ?z) = (yz – y – z, xz – x + z, xy – x + y) = (-6 – 2 + 3 , 3 + 1 – 3 , -2 + 1 + 2) = (-5, 1, 1). pg. 187. Correct answer is C.

The monotonicity property of a function can be investigated by the sign of the derivative. If the
inequality f '(x) > 0 is satisfied for all x ? (a, b) then f(x) is called monotone increasing on the
interval (a, b); likewise, if f '(x) < 0 for all x ? (a, b) then f(x) is called monotone decreasing on
the interval (a, b). The answer is C.

From the product rule for limits, we have, (lim x›2 (5x-2).lim x›2 (3x²+1)) ,so (lim x›2 (5x-2)=8 and lim x›2 (3x²+1)=13 from the product rule again we have, 8*13=104

From the quotient rule for limits together with limx›1(5x-9)?0 we obtain limx›1  2x³+x-12=limx›1(2x³+x-12)/limx›1(5x-9) =-9/-4= 9/4

Since the constant function takes the same value for all x?R, we have

=0,36.

grad f=[(y+yz),(x+2zy+xz),(y²+xy)] at (1,-1,3)=(-4,-2,4)

If a function y = f(x) is differentiable, its derivative f '(x) is a function of x. If the derivative f '(x) is differentiable, the derivative of f '(x) is called the second derivative of y = f(x) and is denoted by y¨=f ¨(x). The derivative of the second derivative, i.e., f ¨(x) is called the third derivative which is denoted by y'''= f ''' (x). For a constant function f(x) = c the derivative is zero, that is, (c)' = 0. The slope of the tangent line to a curve y = f(x) at (x0, f (x0 )) is m = f ' (x0). For the identity function f(x) = x, the derivative is 1, that is, (x)' = 1. The answer is A.

m/n= p/q => m . q = n . p

Therefore, every rational number has an infinite number of representations.

4/5 = 8/10 = 12/15 = 16/20 = ...

at (-1, -2, 1) : ?F / ?x = z = 1 ; ?F / ?z = x + y = 1; ?f / ?x = -(?f / ?x) / (?F / ?z) = - 1 . pg. 190 . Correct answer is B.

P(x) = R(x) – C(x) ; R(x) = (60 – (3 / 4) x) x ; C(x) = 120 + 10 x ; P(x) = -0.75 x² + 50 x – 120 . pg. 161. Correct answer is D.

0 ; Since the degree of the numerator is less than the degree of the denominator. pg. 114. Correct answer is E.

The answer is B.