Math  /  Calculus

QuestionName MP2T2B Mathematics Department Jenlsha Thempson, AP(S) 11/20/202411 / 20 / 2024 Score \qquad Questions 1-13 are 6 points each. Questions 14 is 8 points and question 15 is 14 points. MCs21X - AP Calculus BC Ms. M. E. Dela Cruz
1. Using a GC, find, correct to 3 decimal places, an estimate for the value of xx at which the graph of y=x42x3+5x8y=x^{4}-2 x^{3}+5 x-8 has a horizontal tangent. y=4x36x2+5y^{\prime}=4 x^{3}-6 x^{2}+5
2. Which is true about the function f(x)=x23f(x)=\sqrt[3]{x^{2}} ? (A) It has a vertical tangent at x=0\mathrm{x}=0. (B) It has a stationary point at x=0x=0. (D) It has a cusp at x=0x=0. (C) It has a relative maximum at x=0x=0. (E) It is discontinuous at x=0x=0.
3. If y=xx2+4y=\frac{x}{x^{2}+4}, then dydx=\frac{d y}{d x}= (A) x24(x2+4)2\frac{x^{2}-4}{\left(x^{2}+4\right)^{2}} (B) 4x2(x2+4)2\frac{4-x^{2}}{\left(x^{2}+4\right)^{2}} (1)(x2+4)+(x)(2x)(1)\left(x^{2}+4\right)+(x)(2 x) (x4)+(x1)(x-4)+(x-1)
4. Which of the following functiche (D) 4x2x2+4\frac{4-x^{2}}{x^{2}+4} (E) x24x2+4\frac{x^{2}-4}{x^{2}+4} (A) y=x3y=x^{3} (B) y=1x+1y=\frac{1}{x+1} (C) y=xy=|x| (D) y=xy=\sqrt{x} (E) y=(x2)23y=(x-2)^{\frac{2}{3}}
5. If ff is continuous on [a,b][a, b] which of the following is always true? (A) f\quad f is differentiable on (a,b)(a, b). (B) f\quad f is either increasing or decreasing on [a, b] (C) ff has both a maximum and a minimum value on [a, b]. (D) The maximum value of ff is greater than the minimum value of ff.
6. Find all the intervals over which the function f(x)=x33x2f(x)=x^{3}-3 x^{2} is (a) increasing (6,)(6, \infty) 3x26x3 x^{2}-6 x 6 3x(x26)x=63 x\left(x^{2}-6\right) \quad x=6 (b) decreasing (,0)(0,6)(-\infty, 0) \cup(0,6)
7. Write, in point-slope form, the equation of the tangent line to the graph of y=xx2y=x-x^{2} at (1,0)(1,0).
8. Find the slope-intercept form of the equation of the normal line to the curve y=x3y=x^{3} at the point at which x=13x=\frac{1}{3}.
9. Find the exact values of the absolute extrema of the function f(x)=x3x+2f(x)=\frac{x^{3}}{x+2} on [1,1][-1,1]

Studdy Solution
The value of x x where the graph has a horizontal tangent is approximately x=1.280 x = 1.280 .

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