01:640:135 Study Guide - Final Guide: Riemann SumExam
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Practice ﬁroblems ﬁor Calculus I ﬁnal ﬁxam ﬁall ﬁ008
1. Find the derivative of the function f(x) = cos xsin x.
2. Let F(x) = xex2.
(a) Evaluate Z5
(b) Find the total area of the region bounded by the graph of F(x), the x-axis, x=−2 and x= 5.
(c) Explain the answer for (a) and (b) are diﬀerent.
3. Use diﬀerentials or linear approximation to approximate the value of 2
4. Sketch the graph of the function ln x
You must ﬁnd the domain, all asymptotes, where the function is increasing/decrasing, where it is
concave up/dowm, all local min./max. and all points of inﬂection.
5. Approximate the area of the region that is bounded by the graph of the function f(x) = x2+x
on the interval [0.6], using Riemann sum with 3 equal subintervals. Take the representative points
to be the midpoint of each subinterval.
6. Compute the derivative of Z2
x23etdt twice: directly and then by using the Fundamental Theorem
of Calculus Part 2.
7. Find the function fsuch that f”(x) = sin (2x), f0(0) = 3 and f(0) = 2.
8. Find the dimensions of the largest area rectangle that can be inscribed in the region under the
parabola 4 −x2and above the x-axis.
9. Two circles have the same center. The inner one has radius rwhich is increasing at the rate of
2 inches per second. The outer circle has a radius R which is decreasing at the rate of 3 inches per
second. Let A be the area of the region between the circles. At a certain time, r is 10 inches and
R is 15 inches. How fast is A changing at this time? Is it increasing or decreasing?
10. f(x) is a continuous and diﬀerentiable function on the interval [−2,2]. Here are some values of
the function: f(−2) = 3, f(−1) = 2, f(0) = −3, f(1) = 2 and f(2) = −3.
(a) How many solutions does the equation f(x) = 0 have?
(b) How many solutions does the equation f0(x) = 0 have?
In each (a) and (b) indicate the theorem you are using.
11. Compute the derivative of 1
x−2from the deﬁnition.
12. Evaluate lim
tan 4xtwice: ﬁrst by avoiding L’Hospital Rule and then by using it.
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