Fall Final Review Calc BC Multiple Choice Key PDF

## AP Calculus BC ### 1st Semester Final Review ### Part 1A - No Calculators 1. If $y = (x^2 + 1)^3$, then $\frac{dy}{dx} = ? 3(x^2 + 1)^2.2x = 6x(x^2+1)^2$ Chain 2. If $y = \frac{3x + 4}{4x + 3}$, then $\frac{dy}{dx} = ?$ Quotient $\frac{(3x+4)'(4x+3)-(4x+3)(3x+4)}{(4x+3)^2}$ $\frac{12x+9-12x-1...

## AP Calculus BC ### 1st Semester Final Review ### Part 1A - No Calculators 1. If $y = (x^2 + 1)^3$, then $\frac{dy}{dx} = ? 3(x^2 + 1)^2.2x = 6x(x^2+1)^2$ Chain 2. If $y = \frac{3x + 4}{4x + 3}$, then $\frac{dy}{dx} = ?$ Quotient $\frac{(3x+4)'(4x+3)-(4x+3)(3x+4)}{(4x+3)^2}$ $\frac{12x+9-12x-16}{(4x+3)^2} = \frac{-7}{(4x+3)^2}$ 3. a) $\lim_{x \to 3} \frac{x^2 - 4x + 5}{3x^2 + 9x - 2} = ?$ 3 b) $\lim_{x \to \infty} \frac{x - 4}{3x^2 + 9x - 2} = ?$ 0 c) $\lim_{x \to \infty} \frac{x^3 - 4x +5}{3x^2 + 9x - 2} = ?$ DNE d) $\lim_{x \to 0} \frac{(x + 1)^{\frac{4}{3}} - 1} {x^{\frac{2}{3}}} = \lim_{x \to 0} \frac{(x+1)^{\frac{4}{3}} - 1}{x^{\frac{2}{3}}}\lim_{x \to 0} \frac{\frac{4}{3}(x+1)^{\frac{1} {3}}}{x^{\frac{8} {3}}} = \frac{4}{3}\frac{1}{2} = \frac{2}{3}$ 4. Draw a graph if $f(x) > 0$, $f'(x) < 0$, and $f''(x) > 0$. - above x-axis - decreasing - concave up 5. a) $\lim_{h \to 0} \frac{cos(x+h)-cosx}{h} = ? (cosx)' = -Sinx$ b) $\lim_{h \to 0} \frac{sin(x + h) - sin x}{h} = ? (sinx)' = cosx$ c) $\lim_{h \to 0} \frac{(x+h)^3 - x^3}{h} = ? (x^3)' = 3x^2$ Definition of a Derivative ### Part 1B - Calculators 26. $\frac{d}{dx} \int_{0}^{x^3}sin(t)^2dt = Sin((x^3)^2).3x^2 = Sin(x^6).3x^2$ 27. Let $f(x) = \int_{0}^{x} h(t)dt$, where h has the graph shown above. Draw a possible graph of f(x). 28. The rate of change of the altitude of a hot-air balloon is given by $r(t) = t^3 - 3t^2 + 1$ for $0 < t \le 4$. Write an expression that gives the change in altitude of the balloon during the time the altitude is decreasing. - Decreasing when $r(t) < 0$ 29. If f is a continuous function that is defined for all real numbers with $\int_{0}^{4} f(x)dx = 3$ and $\int_{0}^{4} f(x)dx = 6$ , then $\int_{0}^{4}(f(x) + 2)dx = ?$ $\int_{0}^{4} f(x) dx + \int_{0}^{4} 2dx$ $\int_{0}^{4} f(x) dx - \int_{0}^{4} f(x)dx + 2x|_0^4$ $3-6+2(4)-2(3)$ $3-6-8-6 = -1$ 30. If $g(x) = 2x^2 - 3x + 1$ and $f(x) = g'(x)$, then $\int_{2}^{4} f(x)dx = ?$ $g(4) - g(2)$ $g(4) = 2(4)^2 - 3(4) + 1 = 21$ $g(2) = 2(2)^2 - 3(2) + 1 = 3$ $\frac{21 - 3}{2} = 18$ 31. $\int x cos(3x) dx = \frac{3xSin(3x) + cos(3x)}{3} + c$ 32. Using the substitution $u = \sqrt{x+1}$, $\int_{0}^{3} \frac{x}{ \sqrt{x+1}} dx$ is equivalent to: $\int_{1}^{2} (u^2 -1) 2udu = \int_{1}^{2} (2u^3 - 2u) du$ 33. Let f be the function defined by $f(x) = x^2 + x^3$. If $g(x) = f'(x)$) and $g(2)=1$, what is the value of $g'(2)?$ $g'(x) = \frac{1}{f'(g(x))}$ $g'(2) = \frac{1}{f'(g(2))}$ $f'(x)=4x^2+3x^2$ $f'(1) = 4(1)^2+3(1)^2=7$ 34. A cake heated to a temperature of $350^\circ F$ is taken out of an oven and placed in a $72^\circ F$ room at time $t = 0$ minutes. The temperature of the cake is changing at a rate of $-110e^{-.4t} ^ \circ F$ per minute. To the nearest degree, what is the temperature of the cake at $t = 10$ minutes? $350 + \int_{0}^{10} -110e^{-0.4t}dt = 80^\circ$ 35. The graph of $y = f(x)$ is shown below. If $A_1$ and $A_2$ are positive numbers that represent the areas of the shaded regions, then in terms of $A_1$ and $A_2$, $\int_{-4}^{4} 2f(x)dx - 3 \int_{-1}^{4} f(x) dx = $ $2(A_1 - A_2) - 3(-A_2)$ $2A_1 - 2A_2 + 3A_2 = 2A_1 + A_2$