MATH 131 – Calculus for Physical and Life Sciences Practice Questions for Exam 1 September 17, 2007

Disclaimers: The actual exam questions may be formatted differently and may ask things in different ways, explore different aspects of the topics we have discussed, etc. The following questions only indicate the range of topics that will be covered and the approximate mix of topics and degree of difficulty of the questions on the real exam. There are probably a few more parts here than will appear on the real exam.

I. Let f(x) = \sqrt{1 − x} and g(x) = e^{−x}.

A) What are the functions (f ∘ g)(x) = f(g(x)) and (g ∘ f)(x) = g(f(x))?

B) What is the domain of f ∘ g?

C) Is f ∘ g a one-to-one function? If so, find its inverse function (f ∘ g)^{−1}. If not, say why not.

II. Consider the function K(t), which gives the height (in feet) of a kite above the ground at time t (in seconds):

\begin{matrix}t & 2.0 & 2.5 & 3.0 & 3.5 & 4.0 \\ K(t) & 47 & 54 & 61 & 68 & 75 \end{matrix}

A) Explain why it is reasonable to suspect that K(t) is a linear function, and give a possible formula for K(t).

B) If K(t) is given by your formula from part A, K a one-to-one function. Give a formula for K^{−1}.

C) Using your formula from part A, what is the value K(5) and what does this number represent?

D) What does the number K^{−1}(70) represent.

III. Charcoal is created by burning wood. The amount of radioactive C^{14} left in in the charcoal decays over time. One half of the original amount is left after 5730 years.

A) If 300 grams are present originally, give a formula for the amount left after t years.

B) How much is left after 17190 years?

C) How much is left after 30000 years? Express your answer in exact form, not as a decimal approximation.

IV. Short answer:

A) Find all solutions of the equation | sin(2x)| = 1 for 0 ≤ x ≤ 2π (Note: |x| is the absolute value of x: |x| = x if x ≥ 0 and |x| = −x if x < 0.)

B) Express as a single logarithm value: −\frac{1}{2} ln(x) + 3 ln(x − 1).

C) True or False: For all real x, tan^{−1}(tan(x)) = x. Explain.

D) Simplify: tan(sin^{−1}(3x)). (Hint: Set up a right triangle containing an angle θ = sin^{−1}(3x).)

V. Use graph transformations to sketch the following graphs. Indicate clearly where your graphs cross the x- and y-axes:

A) y = 5 sin(x − π/2) + 3 for 0 ≤ x ≤ 2π B) y = 5 − 2^{x} C) y = 5 − 2^{−x}

VI. The graph y = f(x) and four graphs obtained by transforming it are shown. Match the given formulas with the corresponding graph. Note that there is an extra graph.

−1  ^{0}  ^{1}  ^{2}  ^{3}_{−1}^{1}  y = f(x)
(a) y = f(\frac{1}{2}x) :  (b) y = \frac{1}{2}f(x) :  (c) y = \frac{1}{2} − f(x) :

−1  ^{0}  ^{1}  ^{2}  ^{3}_{−1}^{1}  (i) ^{−1}  ^{0}  ^{1}  ^{2}  ^{3}_{−1}^{1}  (ii)

−1  ^{0}  ^{1}  ^{2}  ^{3}_{−1}^{1}  (iii) ^{−2}  ^{0}  ^{2}  ^{4}  ^{6}_{−1}^{1}  (iv)