AP Calculus AB: Midterm Exam
This set of flashcards focuses on understanding and applying key calculus and algebra concepts such as average rate of change, finding midpoints from intersections, and evaluating limits of functions. The cards help reinforce problem-solving skills with practical examples involving projectiles, intersections, and limits.
A projectile is fired straight up into the air. At 1 second, it has reached a height of 1,584 feet. At 4 seconds, it has reached a height of 6,144 feet. What is the average rate the projectile was traveling in the interval between 1 and 4 seconds?
1,520 ft / s
Key Terms
A projectile is fired straight up into the air. At 1 second, it has reached a height of 1,584 feet. At 4 seconds, it has reached a height of 6,144 feet. What is the average rate the projectile was traveling in the interval between 1 and 4 seconds?
1,520 ft / s
The line of the equation y = x + 1 and the parabola of equation y = x ^2 − 1 intersect at two points, P and Q. What are the coordinates of the midpoint M of the segment PQ ?
M = (1/2, 3/2)
What is the limit of the function in the graph at x = 4?
The limit does not exist.
Determine, if it exists, limx→2 √x+7−3/x^2−4x+4
The limit does not exist.
What is the derivative of the function f (x) = 4x ^3 + 3x ^2 − 2 at x ?
12x^ 2 + 6x
Consider the function y = x^ 2 − x + 7.
At what value of y is the slope of the tangent line equal to 3?
9
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| Term | Definition |
|---|---|
A projectile is fired straight up into the air. At 1 second, it has reached a height of 1,584 feet. At 4 seconds, it has reached a height of 6,144 feet. What is the average rate the projectile was traveling in the interval between 1 and 4 seconds? | 1,520 ft / s |
The line of the equation y = x + 1 and the parabola of equation y = x ^2 − 1 intersect at two points, P and Q. What are the coordinates of the midpoint M of the segment PQ ? | M = (1/2, 3/2) |
What is the limit of the function in the graph at x = 4? | The limit does not exist. |
Determine, if it exists, limx→2 √x+7−3/x^2−4x+4 | The limit does not exist. |
What is the derivative of the function f (x) = 4x ^3 + 3x ^2 − 2 at x ? | 12x^ 2 + 6x |
Consider the function y = x^ 2 − x + 7. At what value of y is the slope of the tangent line equal to 3? | 9 |
Let f′(x)=3x^2+4x define the instanta-neous rate of change (in ft/min) of a car moving along the x-axis. What is the instantaneous rate of change at time 1 min? | 7 ft / min |
What is the derivative of the function f(x)=12/x? | −12/x^2 |
Compute the derivative of the function f(x)=1/(x^2−1)^2 | −4x/(x^2−1)^3 |
Find the derivative of: P(t)=(3t^2/3−6t^1/3)^3 / (3t^2−6t)^1/3 | P′(t)=6(3t^2−6t)^1/3(3t^2/3−6t^1/3)^2(t^−1/3−t^−2/3)/ (3t^2−6t)2^3−2(3t^2/3−6t^1/3)^3(3t^2−6t)^−2/3(t−1)(3t^2−6t)2^3 |
How do you get the graph of sin (2x) from that of sin x ? | Shrink the graph of sin x horizontally by a factor of 2 |
What is the derivative of the function f(x) =sinx(1+e−x)/1+x? | (1+x)[(1+e^−x)cosx+(−e^−x)sinx]−(1+e^−x)sinx/(1+x)^2 |
Evaluate the following as true or false. (ln5)′=1/5 | false |
If dy / dx = 0 for a given value of x, then the line tangent to the curve y = f (x) at that value is horizontal | true |
Find all the points on the curve x ^2 − xy + y^ 2 = 4 where the tangent line has a slope equal to −1. | (2, 2) and (−2, −2) |
Below are the graphs of four functions. Which function is invertible? | For a function to be invertible, it must pass the horizontal line test: any horizontal line must pass through one and only one point on the graph of the function. This is the only graph that satisfies this condition. |
What is the value of d/dx[f−1(x)]when x=2, given that f(x)=x^3+x and f−1(2)=1? | 1/4 |
Evaluate the following as true or false. arccotx=1/arctanx, as long as arccotx and arctanx are both defined. | false |
What is sinh (ln 2) equal to? | 3/4 |
Which of the following is not an equivalent statement of “x approaches c ?” | x = c |