Question
Let g be the function with first derivative \(g'(x)=\sqrt{x^{3}+x}\) for x>0. If g(2)=−7, what is the value of g(5)?
A 4.402
B 11.402
C 13.899
D 20.899
▶️Answer/Explanation
Ans:C
By the Fundamental Theorem of Calculus
$g( 5)−g(2)=\int ^{5}_{2}g'(x)dx$.
Therefore, \(g(5)=g(2)+\int ^{5}_{2}\sqrt{x^{3}+xdx}=-7+\int ^{5}_{2}\sqrt{x^{3}+xdx}=13.899\)
where the evaluation of the definite integral is done with the calculator.
Question
Let
be the function defined by \(f(x)=\frac{1}{4}x^{4}-\frac{2}{3}x^{3}+\frac{1}{2}x^{2}-\frac{1}{2}x\) .For how many values of
in the open interval (0,1.565) is the instantaneous rate of change of
equal to the average rate of change of
on the closed interval [0,1.565] ?
A Zero
B One
C Three
D Four
▶️Answer/Explanation
Ans:C
The average rate of change of
on the closed interval [0,1.565] is \(\frac{f(1.565)-f(0)}{1.565-0}=-0.39206\). The instantaneous rate of change of
is the derivative \(f'(x)=x^{3}-2x^{2}+2-\frac{1}{2}\) .The graph of
, produced using the calculator, intersects the horizontal line y=−0.39206 three times in the open interval (0,1.565).
Question
The function
gives the cost, in dollars, to produce a particular product, where C(x) is the cost, in dollars, to produce x
units of the product. The function
defined by M(x)=C(x+1)−C(x) gives the marginal cost, in dollars, to produce unit number x+1. Which of the following gives the best estimate for the marginal cost, in dollars, to produce the 57th unit of the product?
A \(\frac{C(56)}{56}\)
B \(\frac{C(57)}{57}-\frac{C(56)}{56}\)
C C′(56)
D C′(57)−C′(56)
▶️Answer/Explanation
Ans:C
The marginal cost, in dollars, to produce the 57th unit is given by M(56)=C(56+1)−C(56). This is equal to which is the average rate of change of the cost function C
.
Question
The number of gallons of water in a storage tank at time t, in minutes, is modeled by \(w(t)=25-t^{2}\) for \(0\leq t\leq 5\) .At what rate, in gallons per minute, is the amount of water in the tank changing at time t = 3 minutes?
A 66
B 16
C -3
D -6
▶️Answer/Explanation
Ans:D