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Math 105 Name : Final Exam April 14, 2005
Show all your work. If you use your calculator to compute an answer, you must write down enough information on what you have done that your method is understandable.
- (7 pts.) Use the definition of the derivative as a limit to show that if f(x) = 1 − x^2 , then f′(x) = − 2 x.
- (30 pts. – 3 pts. each) Calculate the following, showing enough steps to make your method clear. Give exact values, not approximations.
(a)
d dx
ln | 1 − 2 cos x + x^2 |
(b)
x +
x +
x
x
dx
(c) F ′(t), for F (t) = xe−x
2 .
(d) d du
7 u^ − 1 u^7 − 1
(e)
d dθ tan^3 (θ + π/4)
(f)
1
x^2 +
x
dx
(g) lim x→ 0
1 − cos x sin(3x)
(h)
− 1
1 + t^2
dt
(i) lim x→∞
x^2 + 1 ex^ + 5x
(j) The slope at the point (− 1 , 1) on the graph of the function y = f(x) defined implicitly by the equation x^3 − 2 xy^2 − 3 y = − 2
- (30 pts. – 2 pts. each) A function y = v(t) is graphed below,and A(x) =
∫ (^) x
0
v(t) dt.
−3 0 1 2 3 4 5 6 7 8 9
−
−
0
1
2
Some of these question are about the function v, and others about the function A. Be careful not to confuse them.
(a) What is the average value of v(t) over the interval [1, 6]?
(b) What is the average rate of change of v(t) over the interval [1, 6]?
(c) What is the instantaneous rate of change of v(t) at t = 3?
(d) A(4) =
(e) A′(5) =
(f) A′′(6) =
(g) For what values, if any, of x ≥ 0 is A(x) decreasing?
(h) For what values, if any. of x ≥ 0 is A(x) concave up?
(i) What x in (0, 9), if any, are critical points of A(x)?
(j) For those x in [1, 6], at what x, if any, does the maximum of A(x) occur?
(k) For those x in [1, 6], at what x, if any, does the minimum of A(x) occur?
(l) At what points in (0, 9), if any, is v(t) not continuous?
(m) At what points in (0, 9), if any, is v(t) not differentiable?
(n) If v(t) described the velocity of an object (in m/sec) at time t (in sec), what is the physical meaning of A(4)? In what units is it measured?
(o) If v(t) described the velocity of an object (in m/sec) at time t (in sec), what is the physical meaning of v′^ (4)? In what units is it measured?
- (6 pts.– 2 pts. each) Let g(x) = x^4.
(a) What, if anything, does the Mean Value Theorem tell you about this function on the interval [− 1 , 2]? Explain.
(b) How do you know that there is some number c for which c^4 = 7? Give an interval in which this c must lie, with justification.
(c) What, if anything, does the Extreme Value theorem tell you about this function on the interval [1, ∞)? Explain.
- (7 pts.) A cell phone tower is located 2 miles off to one side of a highway. How quickly is a car moving away from the tower at the instant when it is already 5 miles from the tower, assuming it then has velocity 65 miles/hr along the road?
