239)
The demand for a product is given by p=200,000
q– 1 . Does the derivative of this function
exist for all values of q?
239)
240)
Differentiate: g(x) = (x– 8)(2x– 7)(x + 5)
240)
241)
Find y’ if y= (x2+ 9) x2+ 4.
241)
242)
One hour after x milligrams of a particular drug are given to a person, the change in body
temperature T(x), in the degrees Fahrenheit, is given approximately by T(x) =x21 –x
11 .
The rate at which T changes with respect to size of dosage x, T’(x), is called the sensitivity of
the body to the dosage. Use the product rule to find the slope of T(x) when the dosage is 7
milligrams.
242)
243)
The total cost (in dollars) of producing x portable radios per day is
C(x) = 1000 + 100x–0.5x2. The marginal cost of producing x radios is found by taking the
derivative of C(x). What is the marginal cost of producing x radios?
243)
244)
A soda stand usually sells 400 sodas per day at $0.70 each. A business student’s research
tells her that for every $0.10 decrease in price, the stand will sell 55 more sodas per day.
The revenue function for the soda stand is given by: R(x) = (0.70 – 0.1x)(400 + 55x) where x
is the number of $0.10 reductions in price. Find dR
dx , the marginal revenue.
244)
245)
Find y’ if y=
3x2+ 8x– 11 7x3– 7x+ 5.
245)
39
246)
Find all values of x for which the curve y=x2+ 6x– 4 has a horizontal tangent line.
246)
247)
If the revenue function is given by R(x) = 6x+ 4, find the derivative of the revenue function
(this is called the marginal revenue).
247)
248)
Suppose a person learns y items in x hours, as given by y= 50 x. Find dy
dx .
248)
249)
The path of a projectile is given by y=x+1
4x2. Find an equation of the tangent to this
curve at x= 2.
249)
250)
The typing speed of a certain computer student varies with the number of hours of practice
x according to S= 21 0.3x. Find S’.
250)
251)
A pizza maker sells large pizzas for $13 each. If he sells q large pizzas, the total revenue r is
given by r= 13q. Find the marginal revenue.
251)
252)
If a manufacturer’s average cost is given by c= 0.001q2– 0.12q+ 7 +7000
q find
(a) the marginal cost function;
(b) evaluate it when 50 units are produced.
252)
253)
The position of an object dropped from a height of 200 feet is given by: y(t) = 200 –16t2.
Find the rate of change of y with respect to t and evaluate it when t= 1 and t= 2. Interpret
your results.
253)
40
254)
Find y if y=3x
5+4
3.
254)
255)
Suppose that the cost C of processing the contaminated water at an industrial site so that
only p percent of the contaminants escape is given by C(p) =750,000 – 7500p
p. Find dC
dp
without using the quotient rule.
255)
256)
If the profit for a telephone company (in millions of dollars) is given by the equation
P(x) = – 11.3x2+ 22x+ 1, use the derivative function on your graphing calculator to find the
slope of the tangent to this curve at x= 3.69.
256)
257)
Find y if y=4x3–6x2+ 7x– 8.
257)
258)
Find y if y=1
4–x
2.
258)
259)
If the velocity function is given by V(x) = 7x2+ 5x+ 1, find an equation of the tangent line
to the graph of V(x) at (0, 1).
259)
260)
Find the slope of the curve y=x + 1 at the point (3, 2).
260)
261)
Find y’ if y=x2
x3+ 4 .
261)
41
262)
The weight W of a tree limb is given by W=3t0.422, where t is time. Find the relative rate
of change of W with respect to t.
262)
263)
Find y if y=2
3x
.
263)
264)
Find y’ if y= (2x+ 1) x– 2.
264)
265)
The path of a projectile is given by y=12x+7x3+ 1
5x. Find an equation of the tangent to this
curve at x= 1.
265)
266)
Find the slope of the curve y=x+ 4 at the point where x= 5.
266)
267)
If the cost C of removing p percent of the particulate pollution from the exhaust gases at an
industrial site is given by C(p) =8100p
100 –p, find C’(p).
267)
268)
True or False: If a function is continuous at x =a, then it is differentiable at x =a.
268)
269)
Find y’ if y=x2(2x–5)5.
269)
270)
Differentiate: f(x) =2x– 9
x + 4
4
270)
271)
If the cost C of removing p percent of the impurities from the waste water in a
manufacturing process is given by C(p) =7700p
100 – 2p, find C’(p).
271)
272)
By direct use of the definition of derivative, find d
dx (f(x)) if f(x) =5
x– 1 .
272)
273)
The demand D for a product at price x is given by D= – 5.2 x+ 22. Find d
dx –5.2 x+ 22
and graph the result on your graphing calculator. What is the behavior of the graph when
x= 0?
273)
274)
A person x inches tall has a pulse rate of y beats per minute, as given approximately by y=
590
x. Find dy
dx .
274)
275)
True or False: If a function is differentiable at x=a, then it is continuous at x =a.
275)
276)
Find the derivative of y =7.1x2– 9.1x+ 2
2.1x2– 1.3x+ 5 .
276)
43
277)
The sales s of a product vary with the number of weeks of an advertising campaign
according to s= 1 +6
4t+ 7 . Use the power rule to find ds
dt .
277)
278)
Differentiate: g(x) =(3x– 9)3
(5x– 7)7
278)
279)
Find the rate of change of y with respect to x when y= 7 – 3x+ 11x3.
279)
280)
Let p= 225 –3q2 be the demand function for a manufacturer’s product. Find the rate of
change of price p per unit with respect to quantity q. How fast is the price changing with
respect to q when q= 10? Assume that p is in dollars.
280)
281)
The demand for a product is given by the equation p=220
q. Find the marginal demand
without using the quotient rule.
281)
282)
Differentiate: f(x) = 2x(x3 + 6x2) – 4(3x2 + x)(8x– 7)
282)
283)
Suppose the position function of an object moving along a line is given by s=f(t) =5t2– 3t
+ 7 where t is in seconds and s is in feet.
(a) Find the average velocity over the interval 5, 5.1 .
(b) Find the velocity when t= 5.
283)
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