Math 131 Fall 2015
Test 2A
Name:
On my honor, as an Aggie, I have neither given nor received
unauthorized aid on this academic work.
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You may use your calculator for all problems on the test but be sure to read each problem
carefully. Some problems ask you to show some algebraic steps.
Below is the graph of the velocity, v(t) of a particle moving left and right on a line. Moving
right is the positive direction. Assume the local max occurs at t = 0, the local min occurs
at t = 2.5, and that v(t) has x-intercepts at t = −1, 2, 3. Let d(t) be it’s position function
and a(t) be it’s accelartion function.
1.
(2 points) At t = 1/2, d(t) is concave up.
True
2.
(2 points) a(0) = 0.
True
3.
False
(2 points) At t = 1 the particle is moving left.
True
4.
False
False
(2 points) d(t) has a horizontal tangent at t = 3.
True
False
5.
(6 points) Which of the graphs below is a possible sketch of the derivative of the above
function.
6.
(6 points) Let f (t) be the temperature at time t where you live. Let f 0 (1) = 2 and
f 00 (1) = 3. Then this means what about the temperature?
(a) The temperature is increasing at an increasing rate.
(b) The temperature is decreasing at an increasing rate.
(c) The temperature is increasing at a decreasing rate.
(d) The temperature is decreasing at a decreasing rate.
7.
(6 points) Given the table of data, estimate f 0 (2.5).
x
2.3 2.4 2.5 2.6 2.7
f (x) 2 2.1 3
4 4.2
(a) Between 3 and 4
(b) Bewteen 5 and 6
(c) Between 6 and 9
(d) Between 9 and 10
8.
√
(10 points) Find the linearization of f (x) =
4.1. Be sure to show the algebraic steps.
9.
√
x − 2 at x = 6 AND use it to approximate
(10 points) Use derivative rules to prove the derivative of cot(x) is − csc2 (x).
10.
(8 points) Find an equation of the tangent line to the curve f (x) = sec(x) at x = π/3.
Give exact answers i.e. leave the square roots as square roots, don’t enter them into your
calculator.
√
Hint: sin(π/3) = 3/2 and cos(π/3) = 1/2.
11.
(10 points) Use definition of the derivative to find the derivative of f (x) =
x
x+1
12.
(12 points) Find f 0 (x) if f (x) = (ecos(x) + cos(ex ))(loga (x2 ))
13.
(12 points) Find f 0 (x) if f (x) = sin
14.
p
(12 points) Find f 0 (x) if f (x) = sin( cos(tan(πx)))
(x+1)2 (x−1)3
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