MIT 18.01 Single Variable Calculus: Practice Final Exam Fall 2006, Study notes of Mathematics

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18.01 Single Variable Calculus

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Fall 2006

18.01 Practice Final Exam

There are 19 problems, totaling 250 points. No books, notes, or calculators. This practice exam should

take 3 hours.

Generally useful trigonometry:

sin^2 x =

1 − cos 2 x ; cos^2 x =

1 + cos 2x ; sec x = ln(sec x + tan x) 2 2

sec x = ; sin

2 x + cos

2 x = 1; tan

2 x + 1 = sec

2 x cos x

In a 30-60-90 right triangle, with hypotenuse 2, the legs are 1 and

3 = 1. 73 � = 3. 14 ln 2 =. 69 ln 10 = 2. 3

Problem 1. (15) Evaluate each of the following:

d ln x a) 2 ; simplify your answer. dx x

b)

d 3 sin

2 u + 2 c)

dn e

kx � � , k constant. du dxn^ x=

Problem 2. (10) Find the equation of the line tangent to the graph of x^2 y^2 + y^3 = 2 at the point

(1, 1) on the graph. (Give the equation in the form y = mx + b.)

Problem 3. (10) Using implicit differentiation, derive the formula for D cos−^1 x by using the

formula for D cos x. (Let y = cos−^1 x.)

x^2 + x + a, x � 0 Problem 4. (10) Let f (x) = , a and b constants. bx + 2, x > 0

Find all values of a and b for which f (x) is differentiable.

Problem 5. (15) On a night when the full moon is directly overhead, an outdoor Christmas tree 50

feet high is falling over. Its top is falling at the rate of 2 feet/sec, at the moment when it is 30 feet from

the ground. At that moment, how rapidly is the shadow of the tree cast by the moon lengthening?

Problem 6. (15) Find the area of the largest rectangle whose base lies along the x-axis and whose

top corners lie on the parabola y = 1 − x^2.

Problem 7. (15: 4,7,4) The graph of y = y(x) has this property: at each point (x, y) on the graph,

the normal line at that point passes through the fixed point (1, 0). (The normal is the line perpendicular

to the tangent line.)

� 1 −^ x a) Show that y = y(x) satisfies the differential equation y =. y

b) Using separation of variables, find all solutions to the differential equation. You can leave the

solutions in implicit form, i.e., as equations connecting x and y.

c) Describe the curves which are their graphs. (You may have to use algebraic processes first (like

completing the square) in order to change the equations into a form where you know what their graphs

look like.)

Problem 8. (15) The cup of a wine-glass has the shape formed by rotating the parabola y = x^2

about the y-axis; its upper rim is a circle of radius 1. How much wine does it hold?