Sample 2 - Math 151 Exam 1 Questions 2 PDF

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Math 151 Exam 1 Questions 2...

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Sample Test 2

( ax2 + x + 1 if x ≤ −1 , find the value of a and b that make f (x) differentiable 1. Consider f (x) = bx − 1 if x > −1 everywhere.

2. Find the derivative. (a) f (x) = ex

2

(b) f (x) = x sin7 (cos(6x))

(c) f (x) =

(x − 1)2 ex2 +2x

1

(d) f (x) = x sec4 (5x)

(e) f (x) = cos(x + e3x )

(f) f (x) =

(4 − x)2 tan x

(g) f (x) = ln(sin2 x)

(h) g(x) = ln(xe−2x )

(i) f (x) = log5 (1 + cos x)

2

(j) f (x) = arcsin(1/x)

(k) f (x) =

√

1 − x2 arcsin x

(l) f (x) = arctan(x2 − x)

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3. Find

dy . dx

(a) x2 y 3 − 5x3 = sec(4y )

(b) tan(xy 2 ) + sin y = 6x2 + 8y + 2

4. If f (x) = cos(ln x2 ), find f ′ (1).

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5. Use the logarithmic differentiation to find the derivative of the function. (a) y =

e−x cos2 x x2 + x + 1

(b) y = xx

(c) y = (ln x)cos x

5

6. Given that h(5) = 3, h′ (5) = −2, g(5) = −3 and g ′(5) = 6, find f ′ (5) for each of the following. (a) f (x) = g(x)h(x)

(b) f (x) =

g(x) h(x)

(c) f (x) = g(h(x))

6

2

7. Find h′′ (1) if h(x) = e−x .

8. Find an unit tangent vector to the curve r(t) =



√ t 2 + 1, t t + t 2



at t = 1.

9. The vector function r(t) = ht + et , t + t2 i represents the position of a particle at time t. Find the speed of the object at the point (1, 0).

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10. At what point on the graph of f (x) =

11. The graph of the curve y = x +

√

x is the tangent line parallel to the line 2x − 3y = 4?

1 cos(3x) has a horizontal tangent at x = 3

8

12. Given f (x) = (1 − x)−1 , find a formula for f (n) (x), the nth derivative of f .

13. The 77th derivative of g(x) = sin(2x).

14. Use a linear approximation at x = 1 to approximate the value of

9

√

1.1.

15. Suppose F and G are differentiable functions. The line y = 1 + 2x is the linear approximation to F (x) F at x = 2, and the line y = 2 − 3x is the linear approximation to G at x = 2. Let H(x) = , G(x) find the linear approximation to H at x = 2.

16. The radius of a circle is measured to be 1 meter with a possible error of ±0.03 m. Use differentials or linear approximation to estimate the maximum possible error in the area of the circle.

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17. A particle moves according to the equation s(t) = t2 − t, where t is measured in seconds and s is in feet. What is the total distance the particle travels during the first 2 seconds?

18. Find the equation of the tangent line to the curve x2 + y 2 = 13 at the point (3, −2).

19. Find the point(s) on the curve x = t2 + 4t, y = t2 + 2t where the tangent line is vertical or horizontal.

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20. Find the equation of the tangent line to the curve parametrized by x = 5t − t3 , y = t2 − 2t at the point corresponding to t = 0.

21. Find the equation of the tangent line to the curve 2x2 y − 3y 2 = −11 at the point (2, −1).

12...