# Pre-assessment ```Name: Moussa Ndiaye
Lesson Title: Investigation
Activity Title: Introduction
Contact Info: [email protected]
Unit #:
1
Lesson #:
1
Date:24-Jul-14
Activity #:
1
Worksheet:
a
Worksheet Title: Pre-Test
Determine the derivative of a function using the limit definition and derivative theorems. Interpret the derivative as
the slope of a tangent line to a graph, the slope of a graph at a point, and the rate of change of a dependent variable
with respect to an independent variable.
1. The graph below shows the concentration C (in moles per liter) of a chemical in a
solution during a chemical reaction as a function of time t (in seconds after the reaction
begins).
a. Estimate the average rate of change in the concentration from t = 5 s to t = 15s.
b. Estimate the instantaneous rate of change in the concentration at t = 20s.
1
Determine the derivative and higher order derivatives of a function explicitly and implicitly and solve related rates
problems.
1. Find formulas for the derivatives of the following functions. Simplify your answers (if
possible).
4
a. y  2
x
b.
y  2 x  1 2
c.
y  ln x 
3
2
d. y  e tan x
e. y 
2
3x  4
x2
2. An object moves as in problem 2a, with displacement, x, given by x  t 3  13t 2  35t  27
where x is in feet and t is in seconds.
a. Find the equation for the velocity and the acceleration.
b. Find the velocity and acceleration at t = 1, t = 6, and t = 8. At each time, state
i. Whether x is increasing or decreasing, and at what rate
ii. Whether the object is speeding up or slowing down, and how you decided
c. At what times in the interval [0, 9] is x a relative maximum? Is x ever negative if t is in
this interval?
3
For problems 1-1, determine a) the intervals on which the function is increasing or decreasing, b)
the x-coordinates of any relative minimum or maximum points on its graph, c) the intervals on
which the function is concave up or concave down, and d) the x-coordinates of any inflection
points on its graph.
1. y  x 3  12 x 2  60 x  10
2. y  ln( x 2  1)
4
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