Math 400 - Final Exam Review
1. Evaluate by any method: lim
a→4
a 3 − 4a 2
a 2 − 16
2. Evaluate by any method: lim
x→3
3. Evaluate by any method: lim
x→∞
4. Evaluate by any method: lim
x→2
x 2 + 5x + 1 − x 2 + 3x + 7
x−3
6e x + 3e− x
2e x + 4e− x
x +1 − x − 5
x 2 + 2x + 3
5. Evaluate by any method: lim
x→2
sin(3x 2 − 12)
ln(x 2 − 3)
6. Evaluate by any method: lim
x→3
2 log(4x − 2)
x 2 + 4x − 6
7. Differentiate f (x) = x 9 sin 2x . DO NOT SIMPLIFY.
x 2 cos 4x
8. Differentiate k(x) =
. DO NOT SIMPLIFY.
sinh 5x
9. Differentiate f (x) =
e5 x + 7
. DO NOT SIMPLIFY.
(x 4 + 1)3/2
10. Differentiate h ( x ) =
e x + x e + e e . DO NOT SIMPLIFY.
(
)
11. Differentiate g(x) = sin ln ( tan(x 7 )) . DO NOT SIMPLIFY.
12. Differentiate f (x) = (2x)2 x . DO NOT SIMPLIFY.
13. Differentiate m(x) = ∫
3x 2
1
2
3t + 7
s2 + 7
ds + ∫ 2
dt . DO NOT SIMPLIFY.
1 t +4
s2 + 4
14. Use the equation below to express
dz
dx
in terms of variables x and z, and constant k if kx 2 + xz + z 2 = k 2
15. Find the point(s) on the parabola y = x 2 closest to the point (1,0).
16. Find and classify all critical values of f (x) = x 6 (1 − x)3 . You do dot need to graph f (x) .
17. Find the absolute maximum and minimum values of f ( x ) = e 2 x ( 2 x − 1) 3 on the closed interval [0,2].
18. Suppose that A(x), B(x), and C(x) are functions and that A = BC . (a) Show that
A′ B′ C ′
=
+
A
B C
(b) Use the
U′
= lnU to deduce the log property: ln(BC) = ln(B) + ln(C)
result from part (a) along with the fact that ⌠
⎮
⌡U
b
19. Using summation notation, write a Reimann Sum approximating the definite integral ∫ x 3dx using n sub0
intervals of equal width and using right end-points as sample points.
20. Evaluate
3
1
∫ 1+ x
2
−
1
21. Evaluate
1
dx .
x3
5
∫
x x − 1 dx
.
2
22. Find the radius and height of the right circular cylinder with volume 18 π cm 3 and least total surface area.
23. Show that f ( x ) =
sin x
∫
cos x
1 − t 2 dt = x −
π
4
f ( x ) = x + C and then show that C = −
24. Find
for all 0 ≤ x ≤ π / 2 (Hint: Show that
π
4
df
= 1, which implies that
dx
.)
dz
where t 2 z 3 + k sin( t 2 ) = k 2 + z 4 (assume that z = z (t ) and k is constant).
dt
8
25. Using n = 4 , find the right-­‐endpoint Riemann Sum approximation of ∫ log 2 x dx . 1
26. Use the product rule to prove the quotient rule. (HINT: Let f (x)
= h(x) , then rewrite this equation g(x)
as f (x) = g(x)⋅ h(x) and differentiate.) 27. Use logarithmic differentiation to find the derivative of x 3 cosh 4 x
e x sin x
. DO NOT SIMPLIFY. x2
28. Let f (x) = 1 +
∫
sin x dx . Find the equation of the line tangent to y = f (x) at x = π / 6 . π 2 /36
29. Show that if f (a + b) = f (a) + bf (a 2 + b 2 ) for all a and b, then df
df
= f (x 2 ) . (Find by the definition) dx
dx
7
30. Using summation notation, find the right end-­‐point Reimann Sum approximation of ∫ x dx . 2
31. Find and classify all critical values of f (x) = x 3 + x 2 − 5x + 7 32. Find the absolute maximum and minimum values of f (x) = x 2 ln x on [1,4e].