Math 1206 Mock Final Spring 2019

Name: _______________________________________________________________________________ Note that both sides of each page may have printed material.

Instructions:

1. Read the instructions. 2. Panic!!! Kidding, don’t panic! I repeat, do NOT panic! Don’t look down, while you’re at it. 3. Complete all problems. 4. You have 135 minutes to complete the test. 5. Show ALL your work to receive full credit. You will get 0 credit for simply writing down the

answers. 6. Crossed-out or erased work will be ignored, even if correct. Problems where several unequal or

contradictory answers are highlighted will be considered incorrect. 7. Write neatly so that I am able to follow your sequence of steps and indicate/box your answers. 8. Read through the exam and complete the problems that are easy (for you) first! 9. Scientific calculators are allowed, but you are NOT allowed to use notes, graphing calculators, or

other aids—including, but not limited to, divine intervention/inspiration, the internet, telepathy, knowledge osmosis, the smart kid that may be sitting beside you or that friend you might be thinking of texting.

10. In fact, cell phones should be out of sight! If you are caught with a cellphone you will be asked to leave the exam and you’ll be given a zero.

11. Use the correct notation and write what you mean! 𝑥2 and 𝑥2 are not the same thing, for example, and I will grade accordingly. Other than that, have fun and good luck!

1. (3 points each) Find the derivative 𝑦′ =𝑑𝑦

𝑑𝑥= for the following.

(a) 𝑦 = 4(2𝑥3 − cos7𝑥)3 +7

√3−𝑥23 .

(b) 𝑦 = tan2 𝑥3 − sin(𝑒𝑥) + 𝑥 sin 𝑥.

(c) 𝑦 = ln√𝑒𝑥

2𝑥+1.

(d) 𝑦 = 𝑥𝑥2+ sec𝑥.

2. (3 points each) Compute the following integrals:

(𝑎) ∫2(𝑥 − 2)2

𝑥𝑑𝑥

(𝑏) ∫(3𝑥 + 1)√3𝑥2 + 2𝑥 − 43

𝑑𝑥

(𝑐) ∫ 𝑥 sin𝑥 𝑑𝑥

(𝑑) ∫3cos𝑥

sin2𝑥 + 1

𝜋/2

0

𝑑𝑥

3. (3 points each) Compute the following limits:

(a) 𝑙𝑖𝑚𝑥→−4

𝑥2+𝑥−12

𝑥+4.

(b) 𝑙𝑖𝑚𝑥→0

𝑥sin3𝑥

sin2𝑥tan3𝑥.

(c) 𝑙𝑖𝑚𝑥→1+

𝑥1/(1−𝑥).

(d) 𝑙𝑖𝑚𝑥→−∞

(√4𝑥 + 𝑥2 + 𝑥).

4. (a) (3 points) Find 𝑑𝑦

𝑑𝑥 given that 𝑥𝑦4 + 𝑥2𝑦 = 𝑥 + 3𝑦.

(b) (2 points) Given that 𝑥𝑦4 + 𝑥2𝑦 = 𝑥 + 3𝑦, find the equation of the tangent line to this

curve at the point (−√3, 1). Write your answer in 𝑦 = 𝑚𝑥 + 𝑏 form.

5. (4 points) Let 𝑓(𝑥) = 𝑥2 +1

1−𝑥. Use the definition of the derivative to find 𝑓′(𝑥). Be sure to

state the definition as a part of your answer. No credit given for any other method. 6. (3 points) Consider the function

𝑔(𝑥) = {𝑥2 − 𝑐2𝑖𝑓𝑥 < 5

𝑐𝑥 + 19𝑖𝑓𝑥 ≥ 5

Find the value(s) of 𝑐 that would make the function 𝑔(𝑥) continuous for all real 𝑥. If no such 𝑐 exist, state so and justify.

7. (5 points) New evidence in the case suggests that Jhevon was dropped off the cliff by his students (allegedly) and he hit the ground at a speed of 160 feet per second. Assuming acceleration due to gravity is -32 feet per second squared, how tall was the cliff? Neglect air resistance and all other factors, treat this as a freefall problem and use calculus to solve it.

8. (a) (4 points) Consider the function 𝑓(𝑥) = 𝑥3 − 3𝑥 + 2 on the interval [−2,2].

Find all numbers 𝑐 that satisfy the conclusion of the Mean Value Theorem on the given interval. In your answer, state how you know the Mean Value Theorem applies.

(b) (4 points) Now consider the function 𝑓(𝑥) = 𝑥3 − 3𝑥 + 2 on the interval [−3,0]. Show that the function has a root in the given interval using the Intermediate Value Theorem. In your answer, conclude why you know there is a root and why the Intermediate Value Theorem applies.

9. (6 points) A circle's area is increasing at a rate of 5 square inches per second. When the area

is 25𝜋 square inches, how fast is the radius changing?

10. (6 points) A rectangle sits in the first quadrant such that its lower left corner is at the origin and its top right corner is touching the circle 𝑥2 + 𝑦2 = 8. What is the area of the largest possible rectangle that can be formed this way?

11. (4 points) Use differentials to approximate the value √8.013

. Write your answer as a single, simplified fraction.

12. (10 points) For the function 𝑓(𝑥) =1−𝑥2

𝑥2−4, you are given (and need not verify) that

𝑓′(𝑥) =6𝑥

(𝑥2−4)2 and 𝑓′′(𝑥) =

−6(3𝑥2+4)

(𝑥2−4)3.

Find the domain, intercept(s), the equations of all asymptote(s), the interval(s) of increase and decrease, local extrema, the interval(s) of concavity, and inflection points (provided these exist). Use this information to sketch the graph.

13. (a) (2 points) Find the average value of the function 𝑓(𝑥) =1

√𝑥 on the interval [1,4].

(b) (2 points) Suppose Find 𝐹′(𝑥). 14. (4 points) Find the absolute extrema of the function 𝑓(𝑥) = 𝑥3 − 6𝑥2 + 9𝑥 + 1 on the interval

[−2,2].

𝐹(𝑥) = ∫1

√𝑡𝑑𝑡

𝑒𝑥

𝑥2+1

.

15. (a) (2 points) Use a Riemann sum with 4 equal subintervals and left-hand endpoints to approximate the area under 𝑦 = 𝑥2 + 1 over the interval [0,8].

(b) (3 points) Find the EXACT area under 𝑦 = 𝑥2 + 1 on the interval [0,8] by taking the limit as 𝑛 → ∞ of a Riemann sum with 𝑛 equal subintervals.