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Math 124

Final Examination

Spring 2023

1. (14 total points) Answer the following.

(a) (5 points) Compute

(b) (4 points) Leibniz computed using L’Hospital’s Rule. What is f(2)? What is f′(2)?

f(2) =                      f ′ (2) =

(c) (5 points) Determine if the following are True or False. Circle the correct answer.

2. (12 total points) Find the derivatives of the following functions. You do not have to simplify your answers.

(a) (4 points)

(b) (4 points) g(x) = (ax2 + b)e−cx where a, b, c are constants.

(c) (4 points) y = (cos(x))x2.

3. (10 points) The function f(x) is differentiable everywhere and f(0) = 0. Answer the questions based on the graph of y = f′(x), the derivative of f(x), shown below.

y = f′(x)

(a)

(b) f′′(11) =

(c) List all values of x where the graph of y = f(x) has a local minimum.

(d) List all intervals where the graph of y = f(x) is concave down.

(e) If g(x) = f (f(x)), what is g′(0)?

4. (12 total points) One point on the curve defined by

x4 + 3xy + y4 = 5

is P = (1, 1).

(a) (4 points) Find the formula for the implicit derivative dy/dx.

(b) (4 points) Write the equation of the tangent line at P.

(c) (4 points) Is the curve defined by the equation concave up or concave down at the point P? Explain.

5. (12 points) A pizza delivery chain has found that the number of veggy pizzas it can sell on a Friday night is closely modeled by the function

N(x) = 1000 − 240 ln(x/12) − 20x,

where x is the cost of the pizza in dollars. The current price is $12. The pizza chain want to have their sales on Friday night increase by 50 pizzas. Use linear approximation to estimate the required new pizza price.

6. (12 points) A balloon is at a height of 40 meters, and is rising at the constant rate of 8m/sec.

At that instant a bicycle passes beneath it, traveling in a straight line at the constant speed of 10m/sec. How fast is the distance between them increasing 3 seconds later?

The picture below is not to scale.

7. (13 points) A tourist group is going to see the ruins across the channel of water. The ruins are at a 100 kilometer horizontal distance from the hotel at a point 10 kilometers across a body of water as shown on the diagram. They can take a bus to drive them by the shore and then they can get on a boat at any point on the shore to get to the other side. The bus travels at 45 kilometers per hour and the boat at 30 kilometers per hour. What is the shortest amount of time in which they can make it to the ruins?

Assume that there is no current and the boat can go straight to the ruins. The drawing is not to scale.

8. (15 points) Let f(x) = 0.5x2e−0.1x be defined on the domain D = [−5, 50].

(a) Determine the sub-intervals of D where f(x) is increasing.

(b) Determine the sub-intervals of D where the graph of y = f(x) is concave up.

8. (continued) Recall the function f(x) = 0.5x2e−0.1x

(c) Find the inflection points on the graph, if any. You can give your answer in decimals rounded to one digit after the decimal.

(d) Find the absolute maximum and absolute minimum values of the function on D. You can give your answer in decimals rounded to one digit after the decimal.

(e) Sketch the graph y = f(x) using the grid below. Clearly label the (x, y) coordinates of endpoints, all critical points, and points of inflection. Make sure your graph matches with the information you provided above.