This

**preview**shows half of the first page. to view the full**2 pages of the document.**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

−2

F(x)dx

(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

√3.99.

4. Sketch the graph of the function ln x

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

x→0

tan 3x

tan 4xtwice: ﬁrst by avoiding L’Hospital Rule and then by using it.

1

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