Derivative Practice Problems PDF

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Derivative Practice Problems (Fall 2019)...

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MATH 1760 Functions and Differential Calculus Problems related to Derivatives 1. Find f (0), where f (x) = ′



x7 cos 0

9 x2

if x 6= 0 if x = 0

2. Use the definition to find the derivatives of the following functions: √ 7 (a) f (x) = 2x2 + 3x − 4 (b) f (x) = x + 8 (c) f (x) = 2 x (d) f (x) = 9x (e) f (x) = cos 5x (f) f (x) = csc x (g) f (x) = sec x 3. Find the following limits. sin2 3x x→0 x

(a) lim

tan 9x x→0 x

x x→0 tan 7x

(b) lim

(c) lim

4. Differentiate the function. You do not need to simplify your result. √ √ 5 4 3 (a) y = x4 + √ − 90 (b) y = x6 − 8 (c) y = (2x2 + 5x)ex x ln x 2 tan x (d) y = ex −7 sin 5x (e) y = (f) y = 7 (x + 2)3 x+1 (g) y = ecos x   1 − e2x (j) y = cot 1 + e2x

(h) y = 7sec x (k) y = 52

x3

5. Find an equation of the tangent line to the curve y =

(i) y = log6 (3x5 − 8) (l) y = √ 3

x csc x e8x − 7

2x − 1 at the point (0, −1).

6. Find the equation of the tangent line to the curve y = x2 ex + tan(x − 1) at the point (1, e). 7. For what values of x does the graph of y = x3 + 3x2 − 70x + 1 have a tangent line with slope 2? 8. At what point on the curve y = e2x +7x is the tangent line parallel to the line 9x−y = 5? 9. For what values of x in [0, 2π], does the graph of y = sin2 x + cos x have a horizontal tangent? 10. Find the points on the curve y = ln(x2 − 16) where the tangent line to the curve is perpendicular to the line 3x + y = 8. 11. Find y ′ . You do not need to simplify your result. (a) x5 + 3xy2 + y 3 = ln y (b) 2x3 y + y 3 = x tan y (c) x2 y = cos(xy) 2

(d) ey/x = 1 + x2 y 3

12. Find an equation of tangent line to the curve x4 − 8y + y 3 = 4 at the point (−1, 3). 13. Find the points on the curve x2 + 4y 2 = 4 where the tangent line is parallel to the line x + 2y = 7 ? 14. Find y ′ . You do not need to simplify your result. (a) y = 5 tan−1 (7x + 2) + csc−1 (4x)

(b) y = x2 sin−1 (x3 )

(c) y = e3x cos−1 (2x2 )

15. Find y ′ . You do not need to simplify your result. (a) y = x2x

(b) y = (cot x)x

2

(c) y = (sin x)x

+2

16. Questions about related rates: From Stewart’s book (8th edition) do examples in Section 3.9 as well as Exercises from Section 3.9 # 3, 13, 15, 17, 19, 21, 23, 29, 31, 43, 45 (7th edition # 3, 11, 13, 15, 17, 19, 21, 27, 29, 39, 41). 17. Use a linear approximation to estimate the following: √ (c) cos 43◦ (a) ln(0.95) (b) 3 27.4...