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Math 220 (section AC2)
Practice Exam 2
Spring 2012
Name
• Do not open this test until I say start.
• Turn off all electronic devices and put away all items except a pen/pencil and an eraser.
• No calculators allowed.
• You must show sufficient work to justify each answer.
• Quit working and close this test booklet when I say stop.
Expectations:
sections 3.1-3.6, 3.11
• Know shortcut derivatives rules for c, x, xn , ex , ax , sin x, cos x, tan x, cot x, sec x, csc x, sin−1 x, cos−1 x,
(x)
(quotient
tan−1 x, ln x, loga x, sinh x, cosh x, cf (x), f (x) ± g(x), f (x)g(x) (product rule), fg(x)
rule), and f (g(x)) chain rule.
• be able to do implicit differentiation (section 3.5)
• be able to take the derivative of f (x)g(x) (section 3.6)
• be able to apply the above derivative rules to problems involving tangent lines, normal
lines, velocity, and acceleration.
• be able to use both Leibniz and prime notation correctly.
section 3.7 Just velocity and acceleration problems
section 3.8 Growth and decay problems. (no Newton’s law of cooling or compound interest.)
section 3.9
• similar triangles
• pythagorean theorem
• trig identities: sin2 x + cos2 x = 1, sin(2x) = 2 sin x cos x
• area formulas for rectangle (length x width), circle (πr2 ), triangle ( 12 bh)
• volume formulas for box (length x width x height), cylinder (πr2 h), any object whose cross
sections are the same (area of base x height)
section 3.10 be able to use linear approximation to approximate given numbers.
section 4.1, 4.3, 4.5
• know the definitions of critical number, increasing, decreasing, concave up, concave down,
inflection point, local max/min, absolute max/min.
• be able to state and use the first derivative test
• given a formula or graph of a function f (x) (or sometimes its derivative f 0 (x)), say where
the graph is increasing, decreasing, concave up, concave down, has an inflection point, has
local max/min, or has abs max/min.
• be able to state the Extreme Value Theorem and use it to find ansolute max/min of a
continuous function on a closed interval.
• be able to sketch a graph using the information from first and second derivatives and
asymptotes.
section 4.2 Be able to state and apply the Mean Value Theorem.
1. Find derivatives of the following functions
(a) f (x) = 2x + 2xex
(b) g(x) =
x5 − 3x
log3 x
√
x + 4x5 + xπ
(c) h(x) =
x
(d) y = sin2 x sin x2 sin2 x2
(e) g(r) =
(f) p(k) =
8−r
ln(r − 4)
q
sin(tan πx)
2. Find f (f 0 (x)) if f (x) = x2 .
3. Find f 0 (x) in terms of g 0 (x) if f (x) = g(x + g(a)) (where you may assume a is a constant.)
4. Use implicit differentiation to find
dy
dx
(a) sin(x − y) = 2x − 2y
(b)
√
x + xy = 1 + x2 y 2
(c) x2 + 2xy − y 2 = 2
5. Find the value of y” at the point where x = 0 if xy + ey = e.
6. Find the tangent line to the graph of the function at the specified input value.
(a) f (x) = xx at x = 2
(b) y = et (cosh t − t2 ) at t = 0
(c) g(x) = cos x + cos 2x at x = π
sin−1 (x)
at x = 1
(d) h(x) =
x
7. The half-life of strontium-90 is 28 days. A sample has a mass of 50 mg initially. Find a
formula for the mass remaining after t days. Find the mass remaining after 20 days. How
long does it take the sample to decay to a mass of 2 mg?
8. (a) State the Mean Value theorem.
(b) State the Extreme Value theorem.
(c) State the definition of a critical number.
9. Any of the assigned related rates problems from 3.9.
10. Approximate (1.999)4 ,
√
99.8, and e−0.015 using linear approximation.
11. Evaluate the following limits. Show sufficient justification. An answer of ”does not exist”
is not sufficient. If the limit is infinite, be sure to state whether it is ∞ or −∞.
(a) lim
x→0
sin 3x
x
cos θ − 1
θ→0
sin θ
(b) lim
12. Sketch a graph of f (x) = 5x2/3 − 2x5/3 (identifying intercepts, asymptotes, intervals of
increase/decrease, intervals of concavity, local extrema, and inflection points.)
√
13. Sketch a graph of x2 + 1−x(identifying intercepts, asymptotes, intervals of increase/decrease,
intervals of concavity, local extrema, and inflection points.)
14. Find absolute and local extrema of
ex
.
1−ex
15. Find the absolute extrema of the function g(x) =
x
x2 −x+1
on the interval [0, 3].
16. Use the Mean Value Theorem to show that there is exactly one solution to the equation
ex = 3 − 2x.