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Name: Student ID: Section: Instructor:
Math 112 (Calculus I)
Final Exam
Dec 18, 7:00 p.m.
Instructions:
- Work on scratch paper will not be graded.
- For questions 11 to 19, show all your work in the space provided.. Full credit will be given only if the necessary work is shown justifying your answer. Please write neatly.
- Should you have need for more space than is alloted to answer a question, use the back of the page the problem is on and indicate this fact.
- Simplify your answers. Expressions such as ln(1), e^0 , sin(π/2), etc. must be simplified for full credit.
- Calculators are not allowed.
For Instructor use only.
Possible Earned
1 10
MC 27
11 7
12 7
13 7
14 7
Sub 65
Possible Earned
15 7
16 7
17 7
18 7
19 7
Sub 35
Total 100
Short Answer Fill in the blank with the appropriate answer.
- (10 points)
a) d dx ln(tan x) =
b) Use the linearization of f (x) = x^1 /^3 at a = 8 to approximate 9^1 /^3.
c) If f ′(x) = x^3 and f (0) = 5 then f (x) =.
d) If f (x) = e^2 x^ then f ′′(x) =.
e) The Mean Value Theorem says that if f is a function on [a, b]
which is also on (a, b) then there is a c ∈ (a, b)
with.
f) Circle the correct answer in both cases: If f ′^ is positive and increasing, then f is (increasing / decreasing) and (concave up / concave down).
g) (^) xlim→∞ 2 x 3 x+3^3 +5x+2 =.
- If f (x) is a differentiable function, which of the following is not always true?
a)
∫ (^) x a
f ′(t)dt = f (x) − f (a) b) d dx
∫ (^) x a
f (t)dt = f (x) c)
∫ f ′(x)dx = f (x) + C
d)
∫ (^) x a
F (t)dt = F ′(x) − F ′(a) e) If∫ F ′(x) = f (x) then x a
f (t)dt = F (x) − F (a)
- A 20 ft ladder is placed against a wall. If the top of the ladder is dropping at a rate of 5 ft/min, how fast is the bottom of the ladder moving away from the wall (in feet per minute) when the top of the ladder is 12 ft high? a) -32 b) -160 c) 160
d)
e)
f)
- If h(x) = x cosh(x^2 − 4), what is h′′(2)? a) 0 b) 4 c) 8
d) 16 e) 32 f) 48
- Let F (x) =
∫ (^2) x 2 3
sin(t) t^3 + 1
dt be defined for x > −1. Find F ′(x).
a) sin(2x^2 ) 8 x^3 + 1 b) 4 x sin(2x^2 ) 8 x^6 + 1 c) sin(2x^2 ) 8 x^3 + 1
sin(18) 216
d) 4 x sin(2x^2 ) 8 x^3 + 1
12 sin(3) 28 e) sin(x) x^3 + 1 f) sin(x) x^3 + 1
sin(3) 28
- If f (x) is a continuous function on the interval [4, 5] and if f (4) = 3 and f (5) = 1, then which theorem guarantees that there is some value c ∈ [4, 5] such that f (c) = 2? a) Mean value Theorem, b) Extreme Value Theorem,
c) Intermediate Value Theorem d) The Fundamental Theorem of Calculus
e) Rolle’s Theorem f) No theorem guarantees this because it is false.
- Use one iteration of Newton’s method, beginning with x 1 = 1/2 to approximate the positive root of the equation x^2 + 2x − 1 = 0. (Note that the root is
a)
b)
c) 0
d)
e)
f) x 2 is undefined.
Free Response. For problems 11 - 19, write your answers in the space provided. Use the back of the page if needed, indicating that fact. Neatly show all work.
- (7 points) State the definition of the derivative of f (t), and use the definition to find the derivative of f (t) = 2t^2 + 1.
- (7 points) Find the equation of the tangent line to y = x
x + 1 at the point (3, 6).
- (7 points) Find the integral
∫ (^3) 1
x 2 x^2 + 1 dx.
- (7 points) An open-at-the-top vertical tank has horizontal cross-section of an equilateral tri- angle, and volume of 4000 cubic feet. Find the dimensions that minimize the surface area. (Hint: the height of an equilateral triangle is
times the length of a side.)
s
h
- (7 points)Given the following graph for f (x), sketch f ′.
- (7 points) Prove that lim x→a 2 x = 2a, using the , δ definition of limit.
END OF EXAM
