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Marywood University Department of Mathematics

Course Information for Math 170

Mathematics for Biology

by Dr. J.

Quartz-filled Dinosaur Bone

The role of mathematics in understanding biology and other life sciences has increased exponentially in recent years. This course studies linear, polynomial, exponential, and logarithmic functions and their uses in modeling biological phenomena. Topics include graphing, limits, continuity, derivatives, techniques of differentiation, and integration. Applications are emphasized. The course is graded based on two tests, weekly group quizes, and a comprehensive final examination. Extra credit is available by presenting homework problems on the board in class. Sample exam problems are given below.

Sample Questions for Test 1

Find the equation of the line that is perpendicular to the line 5x + 10y = 19 and contains the point (-8, 3). [Ans: y = 2x + 19.]
Let f be the function defined by f(x) = 2x^2/sqrt(x-3).
- What is the domain of f?
- Find f(19), f(28), f(t+5)
Sketch these graphs. Be sure to label the intercepts and the asymptotes.
- f(x) = (x+1)/(x^2 - 4)
- g(t) = t^2 -8t + 12
Solve for the exact value of x.
- log_(4)(5x-2) = 3 [Ans: 66/5]
- e^(3x-1) = 7 [Ans: (1+ln7)/3]
When an enzyme is introduced into a culture of 25,000 bacteria, the number of bacteria grows exponentially. After 3.5 hours, there are 60,000 bacteria. How many bacteria will be present after 10 hours (use 4 sig. digits)? [Ans: 304,900]
The half-life of radium-226 is about 1620 years. How much of a sample weighing 2.0 gm will remain after 100 years? [ Ans: 1.916 gm.]
The derivative of f(x) = sqrt(x) is f'(x) = 1/2sqrt(x). Find the equation of the line tangent to the graph at x = 16. [Ans: y = (1/8)x+2.]
Use the limit definition to find the derivative of f(x) = 8/x. [Ans: f'(x) = -8/x^2]

Sample Questions for Test 2

1. Find the derivative of each of the following functions. Do not simplify.

i) f(x) = 3(8x^2 - 2x + 1)^4

ii) g(t) = (t^3 + 7t)/(6t + 1)^4

iii) y = e^(4x) sqrt(x^2 + 5)

iv) h(t) = (2^t) ln(t^2 + 4t^3)

2. Find the equation of the line tangent to y = sqrt(6x - 2) at x = 3. [Ans: y = (3/4)x - 7/4 ]

3. Find f''(0) for i) f(x) = sqrt(x + 4) and ii) f(x) = x ln x.

4. The metabolic rate after a person eats a meal can be modeled by F(t) = -10.28 + 175.9 t e^(-t/1.3) where F(t) is the thermic effect of food and t is the number of hours since eating a meal. Use the first derivative test to find the time after a meal when the thermic effect of food is maximized.

5. Graph y = ln x / x by determining critical points, intervals of increase and decrease, intervals of concavity, and points of inflection.

6. Sketch the graph of a single continuous function that has these properties:

i) f'(x) > 0 on (-infinity, -3) and (1, 4)

ii) f'(x) < 0 on -3, 1) and (4, infinity)

iii) f''(x) > 0 on (-1, 2)

iv) f''(x) < 0 on (-infinity, -1) and (2, infinity)

Sample Questions for Final Exam

1. What is the equation of the line in the xy-plane that goes through the points (-1, 2) and (3, 8)? [Ans: y = (3/2)x + 7/2]

2. What is the domain of each function?

i) f(x) = 3x/(x^2 - 9) [Ans: {x | x ≠ -3, 3} ]

ii) f(x) = x/sqrt(x^2 - 16) [Ans: {x | x < -4 or x > 4}

3. If f(x) = 2x + 1 and g(x) = 3x^2 – x, then g(f(2)) = _____ . [Ans: 70]

4. What is the minimum degree of a polynomial whose graph has 5 turning points? [Ans: 6]

5. Determine the limits:

i) lim (sqrt(x) - 7)/(x - 49) as x --> 49. [Ans: 1/14]

ii) lim (x^2 - x - 6)/(x^2 + 9x + 14) as x --> -2 [Ans: -1]

6. Solve for x: 2^(x+3) = 8^(6x). [Ans: 3/17]

7. Solve for x: log_3(x+1) + log_3(x+7) = 3. [Ans: 2]

8. The annual growth rate of a jellyfish colony in the southern Atlantic is 15%. How long will it take this colony to double? [Ans: 4.6 yrs]

9. Use the definition of limit to find the derivative of f(x) = 5sqrt(x).

10. The half-life of beerillium II is 85 years. How long will it take an original amount of 23.1 grams to decay to 4.6 grams? [Ans: 196.8 yrs]

11. Find these derivatives. Do not simplify.

i) f(t) = (6t^2 + 3t)/(t^3 - 9)

ii) y = 7x^3 - 4sqrt(x^4 + 3x) + e^(8x)

iii) y = (6x + 12)ln(4x^3 + 3x^2)

iv) 7xy^2 + 9x - 11xy = e^x [Ans: dy/dx = (e^x - 7y^2 + 11y - 9)/(14xy - 11x)]

12. Health scientists estimate that days after a particular disease is first observed in the community, the percent of the population infected by the disease is approximated by p(t) = (20t^3 - t^4)/1000 for 0 < t < 20. After how many days is the percent of the population infected a maximum? What is that maximum percent? [Ans: t = 15 days]

13. According to Poiseuille's Law, the velocity v of blood flow in a blood vessel of radius R is v = k(R^2 - r^2), where r is the distance of a layer of blood from the center of the vessel. Suppose a skier has R = 0.08 mm and cold weather is causing the vessel to contract at a rate of dR/dt = -0.01 mm/min. Let k = 375. How fast is the velocity of blood changing for a constant distance r? [Ans: -0.6 mm^3/min.]

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Comments to Dr. Craig M. Johnson, Chair, Dept. of Mathematics: johnsonc@marywood.edu

Last update: December 3, 2007