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Biocalculus Calculus Probability And Statistics For The Life Sciences 1st Edition James Stewart, Troy Day - Solutions
If tsxd − xyex, find tsndsxd.
If f sxd − xex, find the nth derivative, f sndsxd.
If f std − csc t, find f 99s y6d.
If Hs d − sin , find H9s d and H99s d.
y − ex cos x , s0, 1d Find an equation of the tangent line to the given curve at the specified point.37. y − 2xex, s0, 0d Find equations of the tangent line and normal line to the given curve at the specified point.38. y − sx x 1 1 , s4, 0.4d Find equations of the tangent line and normal line
y −2x x 1 1, s1, 1d Find an equation of the tangent line to the given curve at the specified point.
f sxd −ax 1 b cx 1 d Differentiate.
f sxd −x x 1 cx Differentiate.
f sxd −1 2 xex x 1 ex Differentiate.
f sxd −A B 1 Cex Differentiate.
tstd −t 2 st t 1y3 Differentiate.
f std −2t 2 1 st Differentiate.
z − w3y2sw 1 cewd Differentiate.
y −v 3 2 2vsv vDifferentiate.
y −1 2 sec x tan x Differentiate.
y −sin x x2 Differentiate.
y −1 1 sin x x 1 cos x Differentiate.
f s d −sec 1 1 sec Differentiate.
y − sr 2 2 2rder 22. y −1 s 1 kes Differentiate.
y −t st 2 1d2 Differentiate.
y −t 2 1 2 t 4 2 3t 2 1 1 Differentiate.
y −x 1 1 x3 1 x 2 2 Differentiate.
y −x3 1 2 x2 .Differentiate.
f std −2t 4 1 t 2 Differentiate.
tsxd −3x 2 1 2x 1 1 Differentiate.
y −ex 1 1 x Differentiate.
y −ex x2 Differentiate.
y − usa cos u 1 b cot ud Differentiate.
hs d − csc 2 cot Differentiate.
y − 2 csc x 1 5 cos x Differentiate.
f sxd − sin x 1 12 cot x Differentiate.
Rstd − st 1 et d(3 2 st )Differentiate.
Fs yd − S1 y2 2 3y4Ds y 1 5y3d Differentiate.
hstd − et sin t Differentiate.
tstd − t 3 cos t Differentiate.
tsxd − sx ex Differentiate.
f sxd − sx3 1 2xdex Differentiate.
Find the derivative of the function Fsxd −x4 2 5x3 1 sx x2 in two ways: by using the Quotient Rule and by simplifying first. Show that your answers are equivalent. Which method do you prefer?
Find the derivative of f sxd − s1 1 2x2dsx 2 x2d in two ways: by using the Product Rule and by performing the multiplication first. Do your answers agree?
Draw a diagram showing two perpendicular lines that intersect on the y-axis and are both tangent to the parabola y − x2. Where do these lines intersect?
Evaluate lim x l 1 x1000 2 1 x 2 1.
A tangent line is drawn to the hyperbola xy − c at a point P.(a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is P.(b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where P is located
For what values of a and b is the line 2x 1 y − b tangent to the parabola y − ax2 when x − 2?
Find the value of c such that the line y − 32 x 1 6 is tangent to the curve y − csx .
Find the parabola with equation y − ax2 1 bx whose tangent line at (1, 1) has equation y − 3x 2 2.
Prove, using the definition of derivative, that if f sxd − cos x, then f 9sxd − 2sin x.
Use the definition of a derivative to show that if f sxd − 1yx, then f 9sxd − 21yx2. (This proves the Power Rule for the case n − 21.)
(a) Find equations of both lines through the point s2, 23d that are tangent to the parabola y − x2 1 x.(b) Show that there is no line through the point s2, 7d that is tangent to the parabola. Then draw a diagram to see why.
Draw a diagram to show that there are two tangent lines to the parabola y − x2 that pass through the point s0, 24d. Find the coordinates of the points where these tangent lines intersect the parabola.
Where does the normal line to the parabola y − x 2 x2 at the point (1, 0) intersect the parabola a second time?Illustrate with a sketch.
Find an equation of the normal line to the parabola y − x2 2 5x 1 4 that is parallel to the line x 2 3y − 5.
At what point on the curve y − 1 1 2e x 2 3x is the tangent line parallel to the line 3x 2 y − 5? Illustrate by graphing the curve and both lines.
Find equations of both lines that are tangent to the curve y − 1 1 x3 and parallel to the line 12x 2 y − 1.
Find an equation of the tangent line to the curve y − xsx that is parallel to the line y − 1 1 3x.
Show that the curve y − 6x3 1 5x 2 3 has no tangent line with slope 4.
For what values of x does the graph of f sxd − x3 1 3x2 1 x 1 3 have a horizontal tangent?
For what values of x does the graph of f sxd − x 1 2 sin x have a horizontal tangent?
Find the nth derivative of each function by calculating the first few derivatives and observing the pattern that occurs.(a) f sxd − xn (b) f sxd − 1yx
Find d 99 dx99 ssin xd.
(a) Cell growth The volume of a growing spherical cell is V − 43 r 3, where the radius r is measured in micrometers(1 mm − 1026 m). Find the average rate of change of V with respect to r when r changes from(i) 5 to 8 mm (ii) 5 to 6 mm (iii) 5 to 5.1 mm(b) Find the instantaneous rate of change
(a) Find the average rate of change of the area of a circle with respect to its radius r as r changes from(i) 2 to 3 (ii) 2 to 2.5 (iii) 2 to 2.1(b) Find the instantaneous rate of change when r − 2.(c) Show that the rate of change of the area of a circle with respect to its radius (at any r) is
(a) Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate dVydx when x − 3 mm and explain its meaning.(b) Show that the rate of change of the volume
(a) A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 15 mm and it wants to know how the area Asxd of a wafer changes when the side length x changes.Find A9s15d and explain its meaning in this situation.(b) Show that the rate of
If a ball is given a push so that it has an initial velocity of 5 mys rolling down a certain inclined plane, then the distance it has rolled after t seconds is s − 5t 1 3t 2.(a) Find the velocity after 2 s.(b) How long does it take for the velocity to reach 35 mys?
The position function of a particle is given by s − t 3 2 4.5t 2 2 7t, t > 0.(a) Find the velocity and acceleration of the particle.(b) When does the particle reach a velocity of 5 mys?(c) When is the acceleration 0? What is the significance of this value of t?
I nvasive species often display a wave of advance as they colonize new areas. Mathematical models based on random dispersal and reproduction have demonstrated that the speed with which such waves move is given by the expression 2sDr , where r is the reproductive rate of individuals and D is a
Blood flow Refer to the law of laminar flow given in Example 9. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference 3000 dynesycm2, and viscosity − 0.027.(a) Find the velocity of the blood along the centerline r − 0, at radius r − 0.005 cm, and at the wall r − R
R ain forest biodiversity The number of tree species S in a given area A in the Pasoh Forest Reserve in Malaysia has been modeled by the power function SsAd − 0.882A0.842 where A is measured in square meters. Find S9s100d and interpret your answer.
Fish growth Biologists have proposed a cubic polynomial to model the length L of rock bass at age A:L − 0.0155A3 2 0.372A2 1 3.95A 1 1.21 where L is measured in inches and A in years. (See Exercise 1.2.27.) Calculateand interpret your answer. dL dA |A=12
hstd − st 1 5 sin t Find the first and second derivatives of the function.
tstd − 2 cos t 2 3 sin t Find the first and second derivatives of the function.
Gsrd − sr 1 s3 r Find the first and second derivatives of the function.
f sxd − x4 2 3x3 1 16x Find the first and second derivatives of the function.
f sxd − x 1 1x Find f 9sxd. Compare the graphs of f and f 9 and use them to explain why your answer is reasonable.
f sxd − 3x15 2 5x3 1 3 Find f 9sxd. Compare the graphs of f and f 9 and use them to explain why your answer is reasonable.
y − 3x2 2 x3, s1, 2d Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen.
y − x 1 sx , s1, 2d Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen.
y − s1 1 2xd2, s1, 9d Find equations of the tangent line and normal line to the curve at the given point.
y − x4 1 2ex, s0, 2d Find equations of the tangent line and normal line to the curve at the given point.
y − x2 2 x4, s1, 0d Find equations of the tangent line and normal line to the curve at the given point.
y − 6 cos x, s y3, 3d Find equations of the tangent line and normal line to the curve at the given point.
y − x4 1 2x2 2 x, s1, 2d Find an equation of the tangent line to the curve at the given point.
y − s4 x , s1, 1d Find an equation of the tangent line to the curve at the given point.
y − ex11 1 1 Differentiate the function.
Gs yd −A y10 1 Bey Differentiate the function.
v − Ssx 1 1s3 x D2 Differentiate the function.
u − s5 t 1 4st 5 Differentiate the function.
Fsvd − aev 1 bv 1c v 2 Differentiate the function.
f sxd − ksa 2 xdsb 2 xd Differentiate the function.
hsNd − rNS1 2 NKD Differentiate the function.
ts yd −A y10 1 B cos y Differentiate the function.
Ls d −sin 2 1c Differentiate the function.
y − 4 2 Differentiate the function.
tsud − s2 u 1 s3u Differentiate the function.
y −x2 1 4x 1 3 sx Differentiate the function.
f sxd −x2 2 3x 1 1 x2 Differentiate the function.
Fsxd − (12 x)5 Differentiate the function.
y − sx sx 2 1d Differentiate the function.
y − 3ex 1 4s3 x Differentiate the function.
hstd − s4 t 2 4et Differentiate the function.
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