(1) [5%] Find the limit limθ→0cos θ−1 sin θ

MIDTERM 1 FOR CALCULUS

Time: 8:10–9:55 AM, Friday, Nov 3, 2000 Instructor: Shu-Yen Pan

No calculator is allowed. No credit will be given for an answer without reasoning.

1. (1) [5%] Find the limit limθ→0cos θ−1 sin θ . (2) [5%] Find the limit lims→164−√

s s−16. 2. (1) [5%] Find the limit limx→∞

q

2x²−1 x+8x².

(2) [5%] For what value of the constant c is the function f (x) =

( cx + 1, if x ≤ 3;

cx²− 1, if x > 3.

continuous on (−∞, ∞)?

3. (1) [5%] Given the graph of y = f (x) below, sketch a graph of y = f⁰(x).

(2) [5%] Differentiate f (t) = tan(sin t²).

4. (1) [5%] Suppose that u and v are differentiable functions and that w = u ◦ v and u(0) = 1, v(0) = 2, u⁰(0) = 3, u⁰(2) = 4, v⁰(0) = 5, v⁰(2) = 6. Find w⁰(0).

(2) [5%] Show that the curves 3x²+ 2x − 3y²= 1 and 6xy + 2y = 0 are orthogonal.

5. (1) [5%] Find an equation of the tangent line to the curve y = ^√_2−x^|x| ₂ at the point (1, 1).

(2) [5%] Find ^dp_dt if p = (2t − 5)⁴(8t²− 5)⁻³.

6. [10%] Use the linear approximation of the function f (x) =√⁴

x + 1 to estimate √⁴ 1.02.

7. [10%] Find the absolute maximum and absolute minimum values of the function f (x) = _x2^x+1 on the interval [0, 2].

8. [20%] Use the guidelines in the textbook to sketch the graph of the function y = _(x−1)^x 2. 9. [10%] Prove that the equation x³+ 3x + 2 = 0 has exactly one real root.

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