Math 1512 – Final Exam Review – Fall 2024
Topics and sample problems
A. Definitions
Continuity.
1. State the definition of continuity.
2. Let (
GM r/R3 if 0 ≤ r < R
F (r) =
GM/r2 if r ≥ R
where G, M are positive constants. Use the definition to show that F is continuous at r = R.
The derivative.
3. State the definition of the derivative. Sketch a graph that illustrates
4. Use the definition to find the derivative of
√
(a) f (x) = x + 2 (b) f (x) = 1/x2 (c) f (x) = x3
5. True or False: If f is continuous at x = a, then it is differentiable at a. If false, give counterex-
ample.
6. Recognize when a limit is a derivative.
The definite integral.
7. State the definition as a limit of a Riemann sum.
8. Rewriting the limit of a sum as an integral.
n
X √
Evaluate lim xi ∆x, where {xi } is a partition of [0, 1]. (Hint: write as an integral.)
n→∞
i=1
B. Using the Rules
Limits. Find limits as x → c, one-sided limits, infinite limits (vertical asymptotes), limits at infinity.
Including functions whose graphs have holes, functions defined piecewise. Find limits using a graph.
9. Find the following limits. Show all work. Use the symbols ∞ or −∞ if appropriate.
x3 +3x+1 x+1 x−1
(a) lim 3x3 +5
(c) lim x−5 (e) lim 2
x→∞ x→5− x→−2+ x (x+2)
x2 −25 3 2 +3x−1
(b) lim (d) lim x −3xx−1 (f) lim 9−x
√
x→−5 x+5 x→1 x→9 3− x
10. §1.5: 4-9 (find limits and function values using the given graphs)
Derivatives. Use power rule, product rule, quotient rule, chain rule, to find derivatives. Know
derivatives of xp , sin(x), cos(x), tan(x), sec(x), cot(x), csc(x). Know implicit differentiation. Be able
to apply the Fundamental Theorem of Calculus Part 1.
11. Chapter 2 Review Exercises: 13-44 odd
12. Find the derivatives of the following functions.
Z 2p Z 3s
(a) f (x) = x sin(πx) u
(d) f (x) = t2 + 1 dt (g) g(s) = du
x2 2s +1 u2
(b) f (t) = cos(tan t)
(e) f (x) = sec(x3 + 1) E2R
(h) P (R) = , where
√ 1 t (R + r)2
(c) g(s) = s+ √
3 4 (f) s(t) = E, r are constants
s 1 − t2
, 13. Find dy/dx.
p
(a) y = 2x x2 + 1 (c) y = tan2 x (e) sin(xy) = x2 − y 2
3x − 2
(b) y = √ (d) xy 4 + x2 y = x + 3y
2x + 1
14. If f and g are differentiable, find
d hp i d √ d
(a) f (x) (c) xf (x) (e) [f (f (x))]
dx dx dx "s #
d f (x)
d √ d (f)
(b) f ( x) (d) [f (g(x))] dx g(x)
dx dx
15. Let (
GM r/R3 if 0 ≤ r < R
F (r) =
GM/r2 if r ≥ R
where G, M , and R are positive constants. Is F differentiable at r = R?
Indefinite integrals. Find antiderivatives (when possible) either directly or using substitution.
Always check your answer!
16. §3.9: 53, 55 (graphing antiderivative, including of piecewise)
17. §4.5: 9-32 odd
Definite integrals. (Fundamental Theorem of Calculus - Part 2). Find definite integrals. When
using substitution, change the bounds of integration.
18. §4.5: 37-52 odd
19. Chapter 4 Review Exercises: 11-32 odd
Rx
The function g(x) = f (t) dt. What does g(x) represent? What is g 0 (x)? (use FTC - Part 1)
a
R Rx Rb
20. (a) What is the difference between f (x) dx, a f (t) dt, and a f (t) dt?
hR i hR i
d x d x
(b) What is the difference between dx a f (t) dt and dx b f (t) dt , if any?
Rx
21. Find a function f and a value of the constant a such that 2 a f (t) dt = 2 sin x − 1.
Z x2
22. Given the equation x sin(πx) = f (t) dt , find f (4).
0
23. Evaluate
1
Z x
d 1
Z
d 1
(a) dx (c) dt
0 dx 1 + x
2 dx 0 1 + t2
Z 1 Z 1
d 1
d 1
(b) dx (d) dt
dx 0 1 + x2 dx √x 1 + t2
C. Applications of the derivative
Geometric interpretation of the derivative,
24. Chapter 2 Review Exercises: 49-50 (just tangent line), 53, 54 (implicit differentiation)
2
Topics and sample problems
A. Definitions
Continuity.
1. State the definition of continuity.
2. Let (
GM r/R3 if 0 ≤ r < R
F (r) =
GM/r2 if r ≥ R
where G, M are positive constants. Use the definition to show that F is continuous at r = R.
The derivative.
3. State the definition of the derivative. Sketch a graph that illustrates
4. Use the definition to find the derivative of
√
(a) f (x) = x + 2 (b) f (x) = 1/x2 (c) f (x) = x3
5. True or False: If f is continuous at x = a, then it is differentiable at a. If false, give counterex-
ample.
6. Recognize when a limit is a derivative.
The definite integral.
7. State the definition as a limit of a Riemann sum.
8. Rewriting the limit of a sum as an integral.
n
X √
Evaluate lim xi ∆x, where {xi } is a partition of [0, 1]. (Hint: write as an integral.)
n→∞
i=1
B. Using the Rules
Limits. Find limits as x → c, one-sided limits, infinite limits (vertical asymptotes), limits at infinity.
Including functions whose graphs have holes, functions defined piecewise. Find limits using a graph.
9. Find the following limits. Show all work. Use the symbols ∞ or −∞ if appropriate.
x3 +3x+1 x+1 x−1
(a) lim 3x3 +5
(c) lim x−5 (e) lim 2
x→∞ x→5− x→−2+ x (x+2)
x2 −25 3 2 +3x−1
(b) lim (d) lim x −3xx−1 (f) lim 9−x
√
x→−5 x+5 x→1 x→9 3− x
10. §1.5: 4-9 (find limits and function values using the given graphs)
Derivatives. Use power rule, product rule, quotient rule, chain rule, to find derivatives. Know
derivatives of xp , sin(x), cos(x), tan(x), sec(x), cot(x), csc(x). Know implicit differentiation. Be able
to apply the Fundamental Theorem of Calculus Part 1.
11. Chapter 2 Review Exercises: 13-44 odd
12. Find the derivatives of the following functions.
Z 2p Z 3s
(a) f (x) = x sin(πx) u
(d) f (x) = t2 + 1 dt (g) g(s) = du
x2 2s +1 u2
(b) f (t) = cos(tan t)
(e) f (x) = sec(x3 + 1) E2R
(h) P (R) = , where
√ 1 t (R + r)2
(c) g(s) = s+ √
3 4 (f) s(t) = E, r are constants
s 1 − t2
, 13. Find dy/dx.
p
(a) y = 2x x2 + 1 (c) y = tan2 x (e) sin(xy) = x2 − y 2
3x − 2
(b) y = √ (d) xy 4 + x2 y = x + 3y
2x + 1
14. If f and g are differentiable, find
d hp i d √ d
(a) f (x) (c) xf (x) (e) [f (f (x))]
dx dx dx "s #
d f (x)
d √ d (f)
(b) f ( x) (d) [f (g(x))] dx g(x)
dx dx
15. Let (
GM r/R3 if 0 ≤ r < R
F (r) =
GM/r2 if r ≥ R
where G, M , and R are positive constants. Is F differentiable at r = R?
Indefinite integrals. Find antiderivatives (when possible) either directly or using substitution.
Always check your answer!
16. §3.9: 53, 55 (graphing antiderivative, including of piecewise)
17. §4.5: 9-32 odd
Definite integrals. (Fundamental Theorem of Calculus - Part 2). Find definite integrals. When
using substitution, change the bounds of integration.
18. §4.5: 37-52 odd
19. Chapter 4 Review Exercises: 11-32 odd
Rx
The function g(x) = f (t) dt. What does g(x) represent? What is g 0 (x)? (use FTC - Part 1)
a
R Rx Rb
20. (a) What is the difference between f (x) dx, a f (t) dt, and a f (t) dt?
hR i hR i
d x d x
(b) What is the difference between dx a f (t) dt and dx b f (t) dt , if any?
Rx
21. Find a function f and a value of the constant a such that 2 a f (t) dt = 2 sin x − 1.
Z x2
22. Given the equation x sin(πx) = f (t) dt , find f (4).
0
23. Evaluate
1
Z x
d 1
Z
d 1
(a) dx (c) dt
0 dx 1 + x
2 dx 0 1 + t2
Z 1 Z 1
d 1
d 1
(b) dx (d) dt
dx 0 1 + x2 dx √x 1 + t2
C. Applications of the derivative
Geometric interpretation of the derivative,
24. Chapter 2 Review Exercises: 49-50 (just tangent line), 53, 54 (implicit differentiation)
2
