Calculus I Final
1. (a) Let f (x) =
( cx − 2 for x ≤ c;
x2+ 2x for x > c Find c such that f (x) is continuous.
(b) Compute (without the use of L’Hopital’s rule)
x→0lim
√x + 9 − 3 x
2. Find the derivatives of the following functions. It is not necessary to simplify your answer:
(a)
f (x) =
s(3x + 1)2(5x − 1)3 (5x + 2)4 .
(b)
g(x) = x4x
(c)
G(x) = Z 2x
x3
p1 + t4dt
3. Compute (a)
x→0lim
1 − cos 4x 4x2 (b)
x→0lim+(1 + sin x)1/x 4. () Find the asymptotes of
f (x) = (x − 2)2 x2− 4 .
5. An automobile dealer is selling cars at a price of $16,000. The demand function is D(p) = 0.001p(30−0.001p), for 0 < p < 30000, where p is the price of a car. Should the dealer raise or lower the price to increase the revenue? What price will give the dealer maximum revenue?
6. Evaluate the given integral (a)
Z
x(x + 999)7dx, (b)
Z sec2θ
tan2θ − tan θ − 6dθ.
(c)
Z
sec3t dt (d)
Z 1
−1
x−1/3dx
7. Let a curve Γ be defined by: Γ : y =√
4 − 4x2, x ∈ [−1, 1]
(a) Set up a definite integral for the arc length of Γ.
(b) Set up a definite integral for the surface area generated by revolving Γ about x-axis.
8. Solve the IVP, explicitly, if possible y0 = x−1y , y(0) = −2.
9. The plot here shows the relationship between the specific partial pressure of oxygen (pO2, measured in mm Hg) and the saturation level of hemoglobin (y = 1 would mean that no more oxygen can bind). Determine whether f1(x) = x/(27 + x) or f2(x) = x3/(273 + x3) is a better model for this data by finding extrema, inflection points and asymptotes (for x > 0) for each function and comparing to the graph.
