MATH 1220 Midterm: CalcII_Midterm1_Fall2013
1220-90 Exam 1
Fall 2013
Name
Instructions. Show all work and include appropriate explanations when space is provided. Correct
answers unaccompanied by work may not receive full credit. Please circle your final answer. The last page
contains some useful identities.
1. (25pts) Find the indicated derivatives. No need to simplify.
(a) (5pts) Dx(cosh3x)
(b) (5pts) Dx(e5
√x)
(c) (5pts) Dx(ln (x3+ 5))
(d) (5pts) Dx(sin−1(3x+ 5))
(e) (5pts) Dx(xcos x)
1
2. (8pts) Let
f(x) = x3+ 2x+ 1.
(a) (3pts) Why can we conclude that f(x) has an inverse? Hint: Conclude something from f′(x).
(b) (2pts) Fill in the blank: f−1(1) =
(c) (3pts) Compute (f−1)′(1), i.e. compute the derivative of the inverse function to fevaluated at 1.
3. (10pts) Evaluate the following. Any answer representing an angle should be given in radians.
(a) cos−1(√3
2) =
(b) sin−1(0) =
(c) sin (sin−1(1)) =
(d) cos (sin−1(1
7)) =
(e) sin (tan−1(4
5)) =
4. (10pts) A particular radioactive isotope decays from 10 grams to 7 grams in 32 hours. Let A(t) denote
the amount (in grams) of the isotope present after thours.
(a) (6pts) Find constants Cand ksuch that A(t) = Cekt .
C=
k=
(b) (4pts) What is the half-life of this isotope? In other words, how long will it take before half of
the isotope has decayed? No need to simplify.
2
Document Summary
Show all work and include appropriate explanations when space is provided. Instructions. answers unaccompanied by work may not receive full credit. The last page contains some useful identities: (25pts) find the indicated derivatives. No need to simplify. (a) (5pts) dx(cosh3 x) (b) (5pts) dx(e. 5 x) (c) (5pts) dx(ln (x3 + 5)) (d) (5pts) dx(sin 1 (3x + 5)) (e) (5pts) dx(xcos x) Hint: conclude something from f (x). (b) (2pts) fill in the blank: f 1(1) = (c) (3pts) compute (f 1) (1), i. e. compute the derivative of the inverse function to f evaluated at 1: (10pts) evaluate the following. Any answer representing an angle should be given in radians. (a) cos 1( 2 ) = (b) sin 1(0) = (c) sin (sin 1 (1)) = (d) cos (sin 1 ( 1. 7 )) = (e) sin (tan 1 ( 4. 5 )) : (10pts) a particular radioactive isotope decays from 10 grams to 7 grams in 32 hours.
