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August 4, 2014 Name:
Math 1110 – FINAL
- Do not open this booklet until instructed to begin.
- You will have 212 hours to complete the exam. This test has 12 questions, worth a total of 150 points. Please SHOW ALL YOUR WORK : even if your final answer is incorrect you may receive partial credit for the reasoning displayed. Books, notes, calculators, cell phones, and other forms of assistance are not to be used during the exam.
- Each problem appears on its own page and the pages are double sided. There are also two extra sheets of paper at the end for scratch work. Please answer each ques- tion on its page - the scratch work will not be graded.
PROBLEMS GRADES
Total / 150
- (16 pts) Limits Evaluate
(a) Suppose that f(x) is an even function with lim x→ 5 f(x) = 2 , and g(x) is an odd function with lim x→ 5
g(x) = 3. What is lim x→− 5
7f(x)+ 1 g(x)?
(b) lim x→∞
9x^2 − x − 3x) =
- (12 pts) Derivatives
x f(x) f′(x) f′′(x) g(x) g′(x) − 2 6 − 1 − 10 − 4 6 − 1 5 − 2 3 1 0 0 2 − 1 1 4 2 1 0 − 4 8 2 − 4 2 − 1 − 3 2 5 3
Let f and g be differentiable functions where f is one-to-one. Use the above table to determine the derivatives of the following functions at the given points. Note that it is important to take the derivatives first before plugging in the values for x.
(a) h(x) = f−^1 (x) at x = 2
(b) j(x) =
g(x) (t
(^2) + ln t) dt at x = 1
(c) k(x) = 2
x f ′(x) at^ x^ =^0
(d) p(x) = arctan(g(x)) at x = − 2
- (12 pts) Evaluate the following
(a)
∫ (^) π/
0
sin^2 (2θ) cos(2θ) dθ
(b)
x
1 + x dx
(c)
0
4 − x^2 dx
- (10 pts) The acceleration of a particle (moving along a straight line) is given by a(t) = sin( t 2 ) + 6t. Find the position of the particle at time t = π if the initial velocity was v( 0 ) = 5 and the initial position was s( 0 ) = 1. (You do not need to simplify your final answer.)
- (10 pts) Graphing: Consider the function
f(x) = 4x^3 − 24x^2 + 36x = 4x(x − 3 )^2
(a) Find the intervals on which f is increasing and the intervals on which f is de- creasing.
(b) Find the intervals on which f is concave up and the intervals on which f is con- cave down.
(c) Find any local maximums and local minimums and determine which, if any, are global extreme values.
(d) Using what you have calculated in the first four parts, please graph the function
f(x) = 4x^3 − 24x^2 + 36x
on the axes given below. Label any local extreme points, as well as all inflection points.
(c) Let f be a function such that lim x→ 1 +^
f(x) = 3 and suppose f is differentiable at x = 1. Then f( 1 ) must equal 3.
True False
(d) If a marathoner ran the 26.2 mi NYC Marathon in 2 hours, then at least twice the marathoner was running at exactly 11 mph (assuming the initial and final speeds are zero).
True False
- (10 pts) Show that the equation x^5 + 2x^3 + 5x − 3 = 0 has exactly one solution.
- (12 pts) Area:
(a) Draw the region enclosed by the curves y^2 = 4x and y = 2x − 4. Make sure to label your axes and the intercepts.
(b) Find the area of the region enclosed by the curves y^2 = 4x and y = 2x − 4 by integrating with respect to y.
(c) Find the area of the region enclosed by the curves y^2 = 4x and y = 2x − 4 by integrating with respect to x.
(c) Find a closed form for the Riemann sum you found in (b). Note that
∑^ n
k= 1
k =
n(n + 1 ) 2
∑^ n
k= 1
k^2 =
n(n + 1 )(2n + 1 ) 6
(d) Use the answer you found in (c) to determine the actual value for the integral.
(e) Use the Fundamental Theorem of Calculus to determine the actual value for the integral.
SCRATCH WORK
