Math 105A/B Final Exam: Limits, Derivatives, and Integrals, Exams of Calculus

Download Math 105A/B Final Exam: Limits, Derivatives, and Integrals and more Exams Calculus in PDF only on Docsity!

Math 105A/B Final Exam

Name:

• There are 11 questions.
• You have two hours.
• Show all your work.

1. (5 marks) Evaluate the following limits without a calculator. a. limx→ 0 sinx^ x

b. limx→∞ lnx^ x

c. limx→∞ e−^ sin^ x

d. limx→ 1 x x^2 −− 11

e. limx→ 0 x sin (^1) x

3. (5 marks) Find antiderivatives for the following functions. a. f (x) = (^) (3+cossin^ x x) 2

b. f (x) = 3x^2 − 2 √^1 x

c. f (x) = e^2

d. f (x) = − sin xecos^ x

e. f (x) = 5e^2 x

4. (5 marks) Evaluate the following integrals. a. ∫^12 x^2 ln xdx

b. ∫^ −^22 sin xdx

c. ∫^02 x^22 +1x dx

d. ∫^01 e−xdx

e. ∫^0 π (x + cos x)dx

6. (10 marks) Solve the following problems. a. (5 marks) Find the equation of the line tangent to the graph of

f (x) = √x − (^2) x at x = 4.

b. (5 marks) Find the equation of the line tangent to the graph of (x^2 + y^2 )^2 = 4x^2 y at the point x = −1 and y = 1.

7. (10 marks) Solve the following problems. a.[0 (^) , 3]?(5 marks) Where does f (x) = sin x + x 2 achieve a minimum value on

b. (5 marks) Which point on the graph of xy = 4 is closest to the origin?

9. (10 marks) f (x) =

{ sin x 0 ≤ x ≤ π 2 − (^) πx π < x ≤ 2 π Approximate ∫^02 πf (x)dx using the midpoint rule with n = 4.

10. y = (^) −(10 marks) Find the area bounded between the curvesx (^2) + 2. y = x^2 and