Calculus for the Life Sciences By Marvin L. Bittinger
Email: Richard@qwconsultancy.com
Contents Chapter 1 Functions and Graphs…………………………………………….1 Chapter 2 Differentiation…………………………………………………..37 Chapter 3 Applications of Differentiation………………………………….87 Chapter 4 Exponential and Logarithmic Functions……………………….209 Chapter 5 Integration……………………………………………………...253 Chapter 6 Matrices………………………………………………………..323 Chapter 7 Functions of Several Variables………………………………...349 Chapter 8 First-Order Differential Equations……………………………..375 Chapter 9 Higher-Order and Systems of Differential Equations…………409 Chapter 10 Probability……………………………………………………455
Chapter 1
Functions and Graphs 4. A vertical line at x = 10
Exercise Set 1.1
3
1. Graph y = −4. Note that y is constant and therefore any value of x we choose will yield the same value for y, which is −4. Thus, we will have a horizontal line at y = −4.
2 y 1 0
2
4
x 6
8
10
–1
4 –2
y 2
–4
–2
0
–3
2 x
4
–2
5. Graph. Find the slope and the y-intercept of y = −3x. First, we find some points that satisfy the equation, then we plot the ordered pairs and connect the plotted points to get the graph. When x = 0, y = −3(0) = 0, ordered pair (0, 0)
–4
When x = 1, y = −3(1) = −3, ordered pair (1, −3) When x = −1, y = −3(−1) = 3, ordered pair (−1, 3)
2. Horizontal line at y = −3.5
4
4
y 2
y 2
–4
–2
0
2 x
4
–4
4
–4
–4
3. Graph x = −4.5. Note that x is constant and therefore any value of y we choose will yield the same value for x, which is 4.5. Thus, we will have a vertical line at x = −4.5.
Compare the equation y = −3x to the general linear equation form of y = mx + b to conclude the equation has a slope of m = −3 and a y-intercept of (0, 0). 6. Slope
of
m
−0.5
–4
–2
0
2
–2 –1
–4 –2
and
y-intercept
y 2
1
–2
= 4
2
–4 x
2 x
–2
–2
–6
0
–2
2 x
4
of
(0, 0)
2
Chapter 1: Functions and Graphs 7. Graph. Find the slope and the y-intercept of y = 0.5x. First, we find some points that satisfy the equation, then we plot the ordered pairs and connect the plotted points to get the graph. When x = 0, y = 0.5(0) = 0, ordered pair (0, 0)
Compare the equation y = −2x + 3 to the general linear equation form of y = mx + b to conclude the equation has a slope of m = −2 and a y-intercept of (0, 3). 10. Slope of m = −1 and y-intercept of (0, 4) 6
When x = 6, y = 0.5(6) = 3, ordered pair (6, 3) When x = −2, y = 0.5(−2) = −1, ordered pair (−2, −1)
4 y
4
2
y 2
–4
–2
0
–2
2
x
4
6
–2
2 x
4 –4
–2
11. Graph. Find the slope and the y-intercept of y = −x − 2. –4
Compare the equation y = 0.5x to the general linear equation form of y = mx + b to conclude the equation has a slope of m = 0.5 and a y-intercept of (0, 0). 8. Slope of m = 3 and y-intercept of (0, 0)
–4
–2
First, we find some points that satisfy the equation, then we plot the ordered pairs and connect the plotted points to get the graph. When x = 0, y = −(0) − 2 = −2, ordered pair (0, −2) When x = 3, y = −(3) − 2 = −5, ordered pair (3, −5) When x = −2, y = −(−2) − 2 = 0, ordered pair (−2, 0)
4
4
y 2
y 2
0
2 x
4
–4
–2
0
–2
–2
–4
–4
9. Graph. Find the slope and the y-intercept of y = −2x + 3. First, we find some points that satisfy the equation, then we plot the ordered pairs and connect the plotted points to get the graph. When x = 0, y = −2(0) + 3 = 3, ordered pair (0, 3)
2 x
4
Compare the equation y = −x − 2 to the general linear equation form of y = mx + b to conclude the equation has a slope of m = −1 and a y-intercept of (0, −2). 12. Slope of m = −3 and y-intercept of (0, 2)
When x = 2, y = −2(2) + 3 = −1, ordered pair (2, −1)
4
When x = −2, y = −2(−2) + 3 = 7, ordered pair (−2, 7)
y 2
4 –4
y 2
–2
0
2 x
4
–2 –4
–2
0
2 x
4
–4
–2
13. Find the slope and y-intercept of 2x + y − 2 = 0. –4
Exercise Set 1.1
3 22.
Solve the equation for y. 2x + y − 2 y
=
0
y − (−2)
= −2x + 2
Compare to y = mx + b to conclude the equation has a slope of m = −2 and a y-intercept of (0, 2). 14. y = 2x + 3, slope of m = 2 and y-intercept of (0, 3)
y = −3x + 13 23. Find the equation of line: with m = 2, containing (3, 0) Plug the given information into the equation y − y1 = m(x − x1 ) and solve for y
15. Find the slope and y-intercept of 2x + 2y + 5 = 0. Solve the equation for y. 2x + 2y + 5 2y y
=
16. y = x + 2, slope of m = 1 and y-intercept of (0, 2). 17. Find the slope and y-intercept of x = 2y + 8.
1 x−4 2
=
2x − 6
y−0 y
y y
2y
y
= y
18. y = − 14 x + 34 , slope of m = − 14 and y-intercept of (0, 34 )
27. Find the equation of line: with m = 0, containing (2, 3) Plug the given information into the equation y − y1 = m(x − x1 ) and solve for y y − 3 = 0(x − 2) y−3 = 0 y = 3
Plug the given information into equation y−y1 = m(x−x1 ) and solve for y
20. y − 7 = 7(x − 1) y − 7 = 7x − 7 y = 7x 21. Find the equation of line: with m = −2, containing (2, 3) Plug the given information into the equation y − y1 = m(x − x1 ) and solve for y y − 3 = −2(x − 2) y − 3 = −2x + 4 y = −2x + 4 + 3 y = −2x + 7
= mx + b 1 = x + (−6) 2 1 x−6 = 2
26. y = 34 x + 7
19. Find the equation of the line: with m = −5, containing (1, −5)
y − y1 = m(x − x1 ) y − (−5) = −5(x − 1) y + 5 = −5x + 5 y = −5x + 5 − 5 y = −5x
= −5(x − 5) = −5x + 25
Plug the given information into the equation y = mx + b
2y + 8
Compare to y = mx + b to conclude the equation has a slope of m = 12 and a y-intercept of (0, −4).
=
25. Find the equation of line: with y-intercept (0, −6) and m = 12
Solve the equation for y. x−8
2(x − 3)
y 24.
Compare to y = mx + b to conclude the equation has a slope of m = −1 and a y-intercept of (0, − 52 ).
x =
y−0 = 0
= −2x − 5 5 = −x − 2
= −3(x − 5)
y + 2 = −3x + 15
28. y − 8 = 0(x − 4) y−8 = 0 y = 8 29. Find the slope given (−4, −2) and (−2, 1) 1 Use the slope equation m = xy22 −y −x1 . NOTE: It does not matter which point is chosen as (x1 , y1 ) and which is chosen as (x2 , y2 ) as long as the order the point coordinates are subtracted in the same order as illustrated below
m
= = =
1 − (−2) −2 − (−4) 1+2 −2 + 4 3 2