Barrons AP Calculus - David Bock
(C) Let q = (x − 6)^2 + y^2 be the quantity to be minimized. Then q = (x − 6)^2 + (x^2 − 4); q ′ = 0 when x = 3. Note that it s ...
(E) Minimize, if possible, xy, where x^2 + y^2 = 200 (x, y > 0). The derivative of the product is which equals 0 for x = 10. ...
(C) Minimize Since q ′ = 0 if x = 3. Since q ′ is negative to the left of x = 3 and positive to the right, the minimum value o ...
(A) The best approximation for when h is small is the local linear (or tangent line) approximation. If we let then and The appr ...
(A) Since f ′(x) = e−x (1 − x), f ′(0) > 0. ...
(E) The graph shown serves as a counterexample for A−D. ...
(D) Since V = 10w, = 10(8 · −4 + 6 · 2). ...
(E) We differentiate implicitly: 3x^2 + x^2 y ′ + 2xy + 4y ′ = 0. Then At (3, −2), ...
(D) Since ab > 0, a and b have the same sign; therefore f ′′(x) = 12ax^2 + 2b never equals 0. The curve has one horizontal t ...
(C) Since the first derivative is positive, the function must be increasing. However, the negative second derivative indicates ...
(B) Since therefore, at t = 1, Also, x = 3 and y = 2. ...
(A) Let f (x) = x1/3, and find the slope of the tangent line at (64, 4). Since If we move one unit to the left of 64, the tange ...
48. (D) ...
(E) ekh ek·0 + kek·0 (h − 0) = 1 + kh ...
(E) Since the curve has a positive y-intercept, e > 0. Note that f ′(x) = 2cx + d and f ′′(x) = 2c. Since the curve is conca ...
(A) Since the slope of the tangent to the curve is the slope of the normal is ...
(E) The slope at the given point and y = 1. The equation is therefore y − 1 = −1(x + 2) or x + y + 1 = 0. ...
53. (C) ...
(E) Since f ′ < 0 on 5 ≤ x < 7, the function decreases as it approaches the right endpoint. ...
(B) For x < 5, f ′ > 0, so f is increasing; for x > 5, f is decreasing. ...
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