239)
If the velocity function is given by V(x) =7x2– 22.63, find an equation of the tangent line to
the graph of V(x) at (3.51, 63.6107). Use your graphing calculator to verity that the graph of
V(x) and the tangent line are at the point (3.51, 63.6107).
239)
240)
The typing speed of a certain computer student varies with the number of hours of practice
x according to S= 10 0.4x+ 1. Find S’.
240)
241)
Find y if y=1
2–1
2x.
241)
242)
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 .
242)
243)
By using the definition of a derivative, find f‘ (x) where f(x) =2
3x+ 5 .
243)
244)
Money is invested in an annuity that earns r percent per year, compounded monthly. The
future value S of $10,000 invested in such an annuity for 6 years is S= 10,000 1 +0.01r
12
72
.
Find S’.
244)
245)
Find f(u) where f(u) =u5+u11
u3
245)
39
246)
Suppose that the profit p made by selling a certain product at a price x is given by p=f(x)
and the rate of change of that profit with respect to change in price is dp
dx = 8 at x= 3.
Estimate the change in the profit p if the price changes from 3.5 to 4.5.
246)
247)
Find y’ if y=x+ 2
x– 3
4
.
247)
248)
If u= (w+ 1) w+ 1, find the rate of change of u with respect to w when w= 3.
248)
249)
By direct use of the definition of a derivative, find d
dx f(x) if f(x) =1
x+ 6 .
249)
250)
If a manufacturer‘s average cost equation is c= 200 +48q3, find the total cost function c, and
the marginal cost function dc
dq . What is the marginal cost when 17 units are produced?
250)
251)
Find y if y=7x4–5x3+ 4x– 6.
251)
252)
The position of an object thrown upward at a speed of 32 feet/sec from a height of 200 feet
is given by: y(t) = 200 + 32t–16t2. Find the rate of change of y with respect to t and
evaluate it when t= 1 and t= 2. Interpret your results.
252)
40
253)
If a manufacturer‘s average cost equation is c=38
q+ 400 – 25q+11q2, find the total cost
function c, and the marginal cost function dc
dq . What is the marginal cost when 8 units are
produced?
253)
254)
Suppose that the equation r= 430q–2q2 gives the total revenue r (in dollars) that a
manufacturer receives when q units of a product are sold. Determine the marginal revenue
when q= 100 and interpret your result.
254)
255)
A person x inches tall has a pulse rate of y beats per minute, as given approximately by y=
590
x. Find dy
dx .
255)
256)
Suppose that m employees of a company produce a total of q units of a product per day,
where q= 100m–m2, and that the demand equation for the product is p=3000
q+ 100 . Find the
marginal revenue product when m= 10.
256)
257)
The revenue R from the sale of x units of a product is R=5
x+ 50x. The number of units
sold after t weeks of advertising is x= 48 +12
t. Find dR
dt when t= 4.
257)
258)
Find y’ if y= (x2+ 9) x2+ 4.
258)
259)
Suppose that the demand for candy at a price of $p per pound is given by the equation
D(p) = 40 +720
p. Find D(p) when p= 10.
259)
260)
If the velocity function is given by V(x) =9x2+ 12.3, find an equation of the tangent line to
the graph of V(x) at (2.47, 67.2801). Use your graphing calculator to verity that the graph of
V(x) and the tangent line are at the point (2.47, 67.2801).
260)
261)
The demand D for a product at price x is given by D=17
x+ 20. Find d
dx
17
x+ 20 and
graph the result on your graphing calculator. What is the behavior of the graph when x=
0?
261)
262)
Differentiate: f(x) = 2x(x3 + 6x2) – 4(3x2 + x)(8x– 7)
262)
263)
By direct use of the definition of derivative, find d
dx (f(x)) if f(x) =5
x– 1 .
263)
264)
Find y’ if y=1
(2 –3x)4.
264)
265)
Find the derivative of x=f(x) = (1.3t7+9.1t2– 18t+11)(2.1t8+1.8t5– 17t+ 5).
265)
266)
Find y if y=7 – 2x
5
266)
42
267)
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
9.
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 4
267)
268)
True or False: If a function is differentiable at x=a, then it is continuous at x =a.
268)
269)
Money is invested in an annuity that earns r percent per year, compounded monthly. The
future value S of $2000 invested in such an annuity for 5 years is S= 2,000 1 +0.01r
12
60
.
Find S’.
269)
270)
If the cost C of removing p percent of the impurities from the waste water in a
manufacturing process is given by C(p) =9800p
100 – 2p, find C’(p).
270)
271)
Find y’ if y=
3x2+ 8x– 11 7x3– 7x+ 5.
271)
272)
If the velocity function is given by V(x) = 3x–x2, find an equation of the tangent line to the
graph of V(x) at (1, 2).
272)
273)
Find the slope of the curve y=1
x– 10 at the point (11, 1).
273)
274)
An object moves along the ground according to the equation x=t4. Find dx
dt .
274)
275)
Let p= 405 –2q3 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= 4? Assume that p is in dollars.
275)
276)
The demand D for a product at price x is given by D= 8 x. Find d
dx 8x and discuss
what happens at
x= 0.
276)
277)
A cylindrical tank has a height of 7 inches. It is being filled with syrup. Find the rate of
change of its volume with respect to the radius of the cylinder. Evaluate this rate of change
when the radius is 2 inches.
277)
278)
Find y’ if y=2
3 – 4x.
278)
279)
Find the rate of change of y with respect to x when y= 7 – 3x+ 11x3.
279)
280)
Find the rate of change of y=7
x with respect to x when x= 2.
280)
44
281)
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).
281)
282)
Find the equation of the tangent line to the graph of the curve y= 2x2+ 3x+ 11 at the point
(0, 11).
282)
283)
Find y if y=x(x+ 2).
283)
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