University:
Whitman College
Course:
MATH-125 | Calculus I
Academic year:

2021
Views:

409

Pages:

2
Author:

rhitafyw

Calculus I Answers to Sample Exam 1 √ x −2 or explain why it doesn’t exist. x−4 √ √ x −2 x +2 x−4 1 √ √ Answer. lim = lim = lim √ = 1/4. x→4 x − 4 x + 2 x→4 (x − 4)( x + 2) x→4 ( x + 2) 1. Find lim x→4 2. Compute the derivative of f (x) = √ x + 1 directly from the limit definition of the derivative. Do not use any shortcuts. Answer. √ 0 f (x) = lim ∆x→0 x + ∆x + 1 − ∆x √ x+1 √ √ x + ∆x + 1 + x + 1 √ √ x + ∆x + 1 + x + 1 x + ∆x + 1 − (x + 1) √ √ ∆x→0 ∆x( x + ∆x + 1 + x + 1) = lim ∆x √ ∆x→0 ∆x( x + ∆x + 1 + x + 1) 1 √ = lim √ ∆x→0 x + ∆x + 1 + x + 1 1 = √ 2 x+1 = lim 3. Compute √ d 8 (t + 6t7 − 18t2 + 2t − 1/t). dt Answer. 8t7 + 42t6 − 36t + 2 + 1/t2 4. Compute d 3p 3 x x + 3x2 + 5. dx Answer. x3 (1/2)(x3 + 3x2 + 5)−1/2 (3x2 + 6x) + 3x2 5. Compute d 1 . 2 dx (x + 3)2 Answer. −2(x2 + 3)−3 (2x) p x3 + 3x2 + 5 Calculus I—Answers to Sample Exam 1 2 d 5(x2 + 3) √ . dx 15 + x2 √ 2x 15 + x2 − (x2 + 3)(1/2)(15 + x2 )−1/2 (2x) Answer. 5 · 15 + x2 6. Compute d √ ( x + x2 )47 (x2 + 2). dx √ √ Answer. 47( x + x2 )46 ( 12 x−1/2 + 2x)(x2 + 2) + ( x + x2 )47 2x 7. Compute q p d x2 + x3 − x. 8. Compute dx −1/2 −1/2 1 2 p 3 1 3 2 Answer. 2x + x −x (3x − 1) x + x −x 2 2 9. Suppose a ball is thrown straight up so that its height at time t is 30t − 16t2 meters. How fast is the object going at time t = 1? Answer. We want to know the derivative of the function 30t − 16t2 at t = 1. The derivative is 30 − 32t so at t = 1 the ball is going −2 meters per second. 10. Find an equation for the tangent line to f (x) = 4x2 at x = 1. Answer. The derivative is 8x, so when x = 1 the slope of the tangent line is 8. Since f (1) = 4, the tangent line is y − 4 = 8(x − 1) or y = 8x − 4.

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