Calculus
Midterm exam
Nov. 8, 2024
1
(3+3+3+3 points)
Let f (x) =
x2 − 4
x2 + 2x − 3
3
, g(x) =
and h(x) =
. Find the following limits.
2
|x − 2|
x −1
1 + x2
(a) lim f (x)
(b) lim g(x)
x→2
2
x→1
(c) lim h(x)
x→0
(d) lim h ◦ g(x)
x→−1
(2+2+2 points)
If f (x) and g(x) are both continuous functions with lim [3f (x) + g(x)] = 4 and lim [f (x) − 2g(x)] = 6.
x→2
Compute (a) lim f (x)
(b) g(2)
x→2
3
x→2
(c) lim− f (x)g(x)
x→2
(5 points)
If f (x) = x2 + 2x − 3, use the definition of the derivative of a function to compute f 0 (x).
4
(3+3 points)
Use Chain Rule to find the derivatives of the following functions.
(a) f (x) = √
5
2x
x2 + 1
(b) g(x) = sin (2x2 + 3 cos x)
(4 points)
Use implicit differentiation to find dy/dx and d2 y/dx2 of the equation 2xy − 1 = 3x + y 2 .
6
(5 points)
Verify that f (x) = x3 + 2x + 4 satisfies the hypotheses of the Mean Value Theorem on [−1, 1]. Then find all
numbers c that satisfy the conclusion of the Mean Value Theorem.
1
7
(4+3+3+4+6 points)
x4 + x2 + 4x
x
(a) Find the critical points and possible points of inflection for f (x).
(b) Find the open intervals on which f (x) is increasing or decreasing.
(c) Find the open intervals of concavity for f (x).
(d) Find all asymptotes of f (x).
(e) Sketch the graph of f (x), labeling intercepts, relative extrema, points of inflection, and asymptotes.
Let f (x) =
8
(4+4 points)
(a) Find the point on the curve y =
√
x − 8 that is closest to the point (12, 0).
(b) Use Newton’s method with the initial approximation x1 = −1 to find x3 , the third approximation to the
solution of the equation 2x3 − 3x2 + 2 = 0.
9
(5+8+7 points)
(x + 8)
at the point (0, 8).
(a) Find the equation of the tangent line to the graph of f (x) = √
3x + 1
p
(b) Use differentials to approximate 1 + sin (0.01) (by tangent line approximation) .
Z 3
2
√
+ x + 2 dx
(c) Evaluate
3
x
10
(2+2+2+2+3+3 points)
Evaluate the Riemann sum for f (x) = −2x2 + 4x − 1, 0 ≤ x ≤ 2, with eight subintervals, using :
(a) right endpoints as sample points.
(b) left endpoints as sample points.
(c) absolute maximum value of each interval as sample points.
(d) absolute minimun value of each interval as sample points.
Z 2
Z b
n
X
(e) Express the integral
f (x) dx as a limit of sum . [ Hint:
f (x) dx = lim
f (xi )∆x ]
0
a
(f ) Evaluate the limit of (e).
2
n→∞
i=1