Question 1

Suppose $$y = 7x. If$$ $$x$$ increases by $$0.1$$, by how much does $$y$$ increase?

Accepted Answer

$$0.7$$

Question 2

Given $$y = x^2$, what is the slope of the tangent line at x = -3$?

Accepted Answer

$$-6$$

Question 3

Given $$y = x^2$, what is the slope of the tangent line at x = 0$?

Accepted Answer

$$0$$

Question 4

Given $$y = x^2$, what is the slope of the tangent line at x = 2$?

Accepted Answer

$$4$$

Question 5

Refer to the first image (red curve and blue lines). For which values of $$x is$$ $$f'(x) \u003e 2$$?

Accepted Answer

For $$x$$ values where the red curve is steeper than the blue lines (i.e., near the left and right ends of the curve)

Question 6

Refer to the second image (blue curve). For which values of $$x is$$ $$f'(x) = 0$$?

Accepted Answer

At the local maximum and minimum points (approximately $$x \approx 1$$ and $$x \approx 2)$$

Question 7

Refer to the second image (blue curve). For which values of $$x is$$ $$f'(x) \u003c 0$$?

Accepted Answer

Where the curve is decreasing (approximately between $$x = 1$$ and $$x = 2)$$

Question 8

Suppose $$f(3) = 5$$ and $$f'(3) = 2. $$What is the equation of the tangent line to $$y = f(x) at$$ $$x = 3$$?

Accepted Answer

$$y = 2(x - 3) + 5$$

Question 9

If $$f(x) is $$the percentage of children who still get measles when $$x\% of $$children are inoculated, which is a plausible value for $$f'(40)$$?

Accepted Answer

$$-2$$

Question 10

If $$f(40) = 20$$ and $$f'(40) = -2$$, which of the following is true?

Accepted Answer

If the inoculation rate is $$41\%$$, then $$18\% of $$children get measles.

Question 11

Suppose $$T(x) is $$the air temperature in degrees Celsius at a height of $$x$$ meters above sea level. Which of these is a plausible value for $$T'(2000)$$?

Accepted Answer

$$-1$$

Question 12

If $$T(2000) = 10$$ and $$T'(2000) = -1/200$$, which is most plausible?

Accepted Answer

The temperature $$2400$$ meters above sea level is $$8$$ degrees Celsius.

Question 13

Let $$x be $$the amount of money spent in one month on advertising, in thousands of dollars, and $$f(x) is $$the amount of sales during a month, also in thousands of dollars, when $$x is $$spent on advertising. If $$f(20) = 60$$ and $$f'(20) = 3$$, which of the following is true?

Accepted Answer

When the company spends $$20$$ thousand dollars per month on advertising, each extra thousand dollars a month spent on advertising generates an extra $$3$$ thousand dollars of sales.

Question 14

Suppose $$y = f(x) is a $$function and $$f'(a) = 0. $$What does this mean about the graph of $$y = f(x) at$$ $$x = a$$?

Accepted Answer

The tangent line to the graph at $$x = a is $$horizontal.

Question 15

A car's position at time $$t is $$given by $$s(t) = 4t^2$ + 2t. $$What is the car's velocity at $$t = 3$$ seconds?

Accepted Answer

$$26 m/s$$

Question 16

If $$f(x) is $$decreasing for $$x in an $$interval, what can you say about $$f'(x) in $$that interval?

Accepted Answer

$$f'(x) \u003c 0$$

Lecture 17

스터디 가이드 - 연습 문제

학습 가이드 - 플래시카드